WEBVTT

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Game Basics
Mathematical Fundamentals 6 (Dot Product)

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GCC Academy

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Hello everyone

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I am Lee Deuk-woo for gaming maths

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This session focuses on the concept of the dot product

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We’ll discuss

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what a dot product is and how it can be applied

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Dot Product

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The dot product

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is a specialized operation performed on vectors

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Previously, we reviewed basic vector operations

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such as vector addition and scalar multiplication

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These two were the basic operations

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that allow us to create new vectors

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in a vector space through linear combinations

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However, these fundamental operations alone

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may not be sufficient in using vectors

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To address this, mathematicians developed special operations

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that are particularly useful and widely applied

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They are dot and cross products

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Today, we’ll delve into the dot product

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Here are some useful operations for you

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The multiplication of vectors

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The corresponding components are multiplied in the operation

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However, this method is mainly used in contexts like mixing colors

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and is rarely applied to other practical uses of vectors

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On the other hand, the dot product, which we’ll discuss now

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is utilized in almost all applications involving vectors

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Approximately 70–80% of vector-related operations

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begin with the dot product, so it is a very important operation

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Another related operation is the cross product

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This applies exclusively to three-dimensional vectors

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We will discuss the cross product in more detail later

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when we delve into three-dimensional vector operations

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Let’s focus on the dot product

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The dot product, as you can see

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involves two two-dimensional vectors

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multiplying

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the x-components together and the y-components together

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then summing these products

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This process is not limited to two dimensions

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It can extend to three, four

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or even n-dimensional vectors

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Multiply the corresponding components and sum the result

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Regardless of the number of dimensions

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the operation remains consistent

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The dot product involves

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multiplying corresponding components of the vectors and summing them

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which always results in a scalar value

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The properties of the dot product can

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be summarized as follows

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First, the dot product satisfies the commutative property

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Since the operation involves scalar multiplication and addition

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where both multiplication and addition

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are commutative

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the dot product naturally adheres to this property

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Next is the associative property

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The associative property is difficult to apply

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because he operation transitions into scalar-vector multiplication

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It's not an equivalent operation to the original dot product

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The result involves scalar-vector multiplication

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so that is the operation here

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Same thing here, it's scalar-vector multiplication

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So it is not an operation where identical components

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are processed in a sequential manner twice

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For the associative property

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the operation’s nature itself is incompatible,
and the resulting value is fundamentally different

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So, the associative property

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is not applicable to the dot product

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Now, let’s consider the distributive property

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In the case of the distributive property

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if we first calculate the vector addition and then perform the dot product

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the result will be a scalar

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Similarly, if we perform the dot product on each term individually

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and then sum the scalar results

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the outcome will be the same

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This demonstrates that the dot product

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satisfies the distributive property

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To summarize

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the dot product satisfies the commutative property
but does not satisfy the associative property

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However, it does satisfy the distributive property

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Now, if we examine the characteristics of the dot product

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through the lens of the distributive property

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The dot product applied to the addition of the same vectors

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We get terms like u.u, and u.v

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v.u and v.v

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Due to the commutative property, u.v and v.u are equal

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which simplifies the expression to u.u, v.v, e.u.v

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When we calculate the dot product

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of the same vector with itself

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as seen here, it results to where

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each component is squared

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For example, the first component is squared

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the second component is squared, and so on

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That's what we get

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This operation ultimately corresponds

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to the square of the norm of the vector

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Therefore, taking the dot product of a vector with itself

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always results in the square of its norm

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or the square of its length

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Thus, the equation above can be rewritten

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in terms of squared magnitudes

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the square of the norm of vector u

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the square of the norm of vector v

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and the interaction term e.u.v

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This equation is formed in this way

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and since it will prove useful later

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I wanted to mention it here

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To summarize

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the norm of vector v is calculated

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as the square root of the sum of the squares of its components

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When the same vector is involved in the dot product with itself,
the result is always the square of this norm

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For any n -dimensional vector

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the dot product of the vector with itself yields the square of its norm

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That's the conclusion we get

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Using these properties

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the dot product can be applied in various ways

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First, it’s important to understand the properties of the dot product

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which are closely related to trigonometric functions

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particularly the cosine function

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In fact, the reason the dot product

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is so widely used in game production

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in its application, in particular

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because it relates to the cosine function

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By leveraging the properties of the cosine function

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the dot product can be utilized in a variety of situations

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So, the formula for that is as follows

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The dot product of a and b

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is equal to the product of the magnitude of vector a and that of b

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and the cosine of the angle theta between them

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This formula is what enables this

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To understand how this formula is derived, let’s briefly examine it

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First, imagine an arbitrary triangle with points

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point A, B, and C

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Place point B at the origin

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Define the vector pointing from B to A  as vector C

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and the vector pointing from B to C as vector a

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so the side opposite point A is referred to as vector a

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So the side opposite to that is vector b

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When arranging the triangle in this way

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we can use this triangle

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to derive the cosine formula over here

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Point B is set at the origin, two sides as vectors

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so b defined as (0, 0)

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Point C is positioned to align with the x-axis

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The side opposite point A is defined as vector a

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If we set vector a in the positive direction

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its magnitude

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corresponds to the norm of vector a

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So it's the length

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So it's vector a norm, 0

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Now, for point A

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assume the angle

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between vectors c and a is beta

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To determine the height

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we refer to unit circle properties

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For a vector with a magnitude of 1

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the x-component is cosine theta

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and the y-component is sine theta

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If the radius is r, the components become r cosine theta and r sine theta

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With vector c, whose magnitude is known

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and the angle beta

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it's the norm of c times cosine beta for the x-coordinate

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and the norm of c times sine beta for the y-coordinate

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If the positions of the points are defined as stated

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we can calculate vector b

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Vector b is pointing from point C to point A

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or equivalently, from point A to point C

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The coordinates of vector b can be determined based on earlier definitions

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Next, we calculate the magnitude of vector b

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and then examine the result of squaring this magnitude

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Squaring each component of vector b results in an expanded form

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The sum of the square of the norm of vector a, that of vector c

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and twice the product of the norms of vectors a and c

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multiplied by the cosine of the angle beta between them

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When the magnitude of vector b is squared

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it is equivalent to taking the dot product of vector b with itself

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Thus, this result represents b.b

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which can also be expressed in terms of vectors a and c

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by rewriting the relationship

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So that's the dot product

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Specifically, the result becomes

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the sum of the square of the norm of vector a, that of vector c

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and -2a.c, where a.c is the dot product of vectors a and c

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So this value is same as the one over here

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From here, by canceling out

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the square of the norm of a and the square

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of the norm of c

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we find that the dot product a.c

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equals the product of the magnitudes of vectors a and c

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multiplied by the cosine of the angle beta between them

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So this is the conclusion we get

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Using the dot product

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we derived the cosine formula from trigonometric principles

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If the magnitudes of the two vectors

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are both 1, the formula simplifies further

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reducing to the cosine of the angle

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between the two vectors, the cosine of theta

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This property is highly useful in various applications

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We can effectively utilize proportional relationship

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between the dot product and the cosine function

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The cosine function’s properties can be applied in various ways

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with one of the most notable applications

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being orthogonality testing

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This determines whether two vectors are perpendicular to each other

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If the dot product of two vectors equals 0

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it implies that the cosine of the angle theta between them must also be 0

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For this to happen

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if not 0

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the cosine of theta can only be 0

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for this value to be 0

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So when does the cosine of theta become 0?

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90 and 270 degrees

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It's when the angle is 90 and -90 degrees

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The condition for the dot product to be 0 directly

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translates into the vectors being orthogonal

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By leveraging this property of the dot product

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we can tell

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that the dot product of two vectors is 0

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and it happens when

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the two vectors are perpendicular

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If we calculate the dot product of the standard basis vectors 1, 0 and 0, 1

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1 by 0 by 0 by 1

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it's a 0

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This result indicates that the vectors are orthogonal

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If we then rotate these vectors by an angle theta

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so the perpendicular vector is rotated by theta

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so the rotated vectors remain orthogonal

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However, instead of relying on visual intuition

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we can simply calculate the dot product of the rotated vectors

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If the rotated vectors are represented as (cosine theta, sine theta)

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and (negative sine theta, cosine theta)

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their dot product becomes

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cosine theta times negative sine theta

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plus sine theta times cosine theta, which is 0

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This confirms that

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the two rotated vectors are also orthogonal

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Building on this concept

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we can introduce the idea of a rigid transformation

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So

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rigid means firm or solid

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and in physics, it refers to a rigid body

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which is a solid object whose shape does not deform

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A rigid body

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It's rigid body in physics

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It can be thought of in a similar way

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as in game development

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Rigid transformation refers to

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a type of transformation where the object’s shape

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remains unchanged during the transformation

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Rigid transformation

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For a transformation to qualify as a rigid transformation,
the following must be satisfied

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After the transformation, the magnitude of all basis vectors must remain 1

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All basis vectors must remain orthogonal to each other

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because the original vectors in the space are orthogonal

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and this property must be preserved after the transformation

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The determinant of the transformation matrix must be 1

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to ensure that the object does not flip

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To confirm that a transformation is rigid

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we can verify that the determinant of the transformation matrix is 1

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Rigid transformation

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It ensures that the shape of the object

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remains completely unchanged

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A prime example of a rigid transformation is rotation

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When examining rotational transformations

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we see that the magnitude of each basis vector

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in the transformed space remains 1

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Let's consider the basis vector represented by cosine theta and sine theta

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Its norm is given by the square root of 
cosine squared theta plus sine squared theta

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According to the Pythagorean theorem

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we know this equals 1

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When squared, the negative sign in the second basis vector disappears

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resulting in cosine squared theta plus sine squared theta

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The second basis vector also has a magnitude of 1

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It hold true regardless of whether we examine 
the vectors horizontally or vertically

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We also previously verified that the basis vectors

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remain orthogonal to each other

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Next, let’s examine the determinant of the transformation matrix

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ad-bc

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In this case, cosine squared theta plus sine squared theta

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is equal to the determinant value ad - bc

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This confirms that the determinant of the matrix is 1

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00:18:05.119 --> 00:18:07.220
Such a rotational transformation

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ensures that the shape and structure of an object remain

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unchanged after the transformation

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This is the rigid transformation

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Application of the Dot Product

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Let’s examine how the dot product is practically applied

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One prominent application is determining the front or the back

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00:18:33.480 --> 00:18:36.919
By using the cosine formula embedded in the dot product

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we can evaluate the relative position of a target

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For example, imagine a person facing

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forward with a view vector

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If there is an object in space

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we can define a vector

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pointing from the person to the object

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When the angle between

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these two vectors is theta

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taking their dot product produces a value proportional to cosine theta

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00:19:07.479 --> 00:19:09.120
The graph of cosine

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00:19:09.121 --> 00:19:11.943
starts at 1 when x equals 0

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00:19:11.944 --> 00:19:16.521
and decreases as the angle moves

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00:19:16.521 --> 00:19:18.278
How does it work?

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Toward 90 degrees or negative 90 degrees

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00:19:22.532 --> 00:19:23.839
reaching 0 at these limits

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00:19:23.840 --> 00:19:26.480
beyond this, the cosine value turns negative

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00:19:26.834 --> 00:19:32.761
This means that the values are positive from -90 degrees to 90 degrees

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00:19:32.762 --> 00:19:35.984
As long as the angle between the two vectors

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the values remain positive until they reach plus and minus 90 degrees

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Beyond this range, the cosine value becomes negative

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Since the magnitude of a vector is never negative

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when the dot product of two vectors yields a positive result

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00:19:52.319 --> 00:19:55.680
it indicates that the cosine of the angle is positive

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00:19:55.924 --> 00:19:59.903
Conversely, if the dot product is negative,
it signifies that the cosine of the angle is negative

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00:20:00.319 --> 00:20:03.964
Therefore, if the dot product of the two vectors is positive

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00:20:04.469 --> 00:20:08.109
it implies that the object is on the same side

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as the viewing direction

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which we describe as being in front

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00:20:12.529 --> 00:20:16.540
On the other hand, if the dot product is negative

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00:20:16.540 --> 00:20:18.829
it indicates that the object is on the opposite side, or behind

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00:20:19.680 --> 00:20:22.739
This concept can be further applied

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to determine whether an object lies within a specific field of view

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00:20:26.439 --> 00:20:29.440
Consider the following scenario

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Based on the direction you’re facing

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the angle between your line of sight and the target object is called alpha

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00:20:39.673 --> 00:20:44.800
You can use this to check

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00:20:45.301 --> 00:20:49.400
whether alpha falls within

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00:20:49.400 --> 00:20:51.987
a predefined viewing angle, beta

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00:20:52.119 --> 00:20:57.023
The field of view is typically defined symmetrically

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around the viewing vector

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so half of the field of view on one side is beta divided by 2

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00:21:05.368 --> 00:21:09.637
The task is to determine whether the target lies

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00:21:09.638 --> 00:21:14.172
within this range, meaning whether alpha is

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00:21:14.172 --> 00:21:15.929
less than beta divided by 2

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00:21:16.213 --> 00:21:21.121
If the object is detected within the field of view,
it means alpha is smaller than beta divided by 2

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00:21:21.598 --> 00:21:25.298
However, while the field of view beta is known

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00:21:25.299 --> 00:21:27.894
the specific value of alpha may not be readily available

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00:21:27.894 --> 00:21:28.760
and needs to be calculated

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00:21:29.134 --> 00:21:32.074
To measure the angle

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00:21:32.074 --> 00:21:37.919
one could use inverse functions such as the arctangent

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00:21:38.259 --> 00:21:40.322
However, using the dot product is a more efficient way

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00:21:40.322 --> 00:21:43.956
to determine whether an object lies

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00:21:43.957 --> 00:21:46.444
within the field of view without needing

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00:21:46.757 --> 00:21:48.817
to calculate the actual angle alpha

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00:21:50.520 --> 00:21:52.110
Here’s how it works

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First, precompute and store the value of cosine of beta divided by 2

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00:21:55.841 --> 00:21:57.439
Since the field of view

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00:21:58.822 --> 00:22:02.782
for a character typically doesn’t change often

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00:22:03.040 --> 00:22:07.834
precomputing and storing this value is advantageous

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00:22:08.312 --> 00:22:11.352
Next, normalize both the viewing vector and

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00:22:11.353 --> 00:22:13.658
the vector pointing toward the target

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00:22:13.658 --> 00:22:16.873
Normalization ensures that both vectors have a magnitude of 1

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00:22:17.488 --> 00:22:22.381
When these normalized vectors are dot-multiplied

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the result is the cosine of alpha, the angle between them

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00:22:25.112 --> 00:22:28.111
Due to the properties of the cosine function

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00:22:28.872 --> 00:22:31.672
its value decreases as the angle increases from 0

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00:22:31.672 --> 00:22:34.640
all the way to 180 degrees

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All the way down

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The cosine function decreases in value as the angle increases

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00:22:42.199 --> 00:22:46.190
Therefore, if the cosine of alpha is greater than

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00:22:47.593 --> 00:22:52.372
the cosine of beta divided by 2

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00:22:52.373 --> 00:22:54.933
it means the angle alpha is smaller than that

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00:22:55.191 --> 00:22:57.644
By comparing the two cosine values

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00:22:57.645 --> 00:23:01.651
if the cosine of alpha is greater than the cosine of beta divided by 2

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00:23:02.322 --> 00:23:05.942
we can determine that the target is within the field of view

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00:23:06.336 --> 00:23:09.356
even without calculating the exact angle

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00:23:09.800 --> 00:23:15.443
While inverse functions can sometimes be used

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00:23:15.443 --> 00:23:19.352
leveraging the properties of the dot product

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00:23:19.353 --> 00:23:23.492
is far more efficient and practical for this purpose

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00:23:24.492 --> 00:23:27.664
Next, let’s look at shading calculations

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In computer graphics, particularly in 3D rendering

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the direction of light significantly

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influences shading

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The dot product is an essential tool here

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00:23:43.241 --> 00:23:46.054
When there is a light source

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00:23:48.120 --> 00:23:52.581
the light strikes a surface, reflects off it

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00:23:53.743 --> 00:23:56.523
and eventually reaches our eyes

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00:23:56.900 --> 00:24:01.460
The amount of light illuminating the surface

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00:24:01.923 --> 00:24:07.823
is maximized when the surface is oriented perpendicular

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00:24:08.479 --> 00:24:10.302
to the direction of the incoming light

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00:24:10.895 --> 00:24:15.834
As the angle between the light direction

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00:24:17.459 --> 00:24:19.719
and the surface normal increases

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00:24:19.720 --> 00:24:23.595
the amount of incoming light gradually decreases

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00:24:23.595 --> 00:24:25.559
This decrease is

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00:24:25.559 --> 00:24:32.458
directly proportional to the cosine of the angle 
between them, described in Lambert’s Cosine Law

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00:24:32.458 --> 00:24:37.998
Shading is determined based on the cosine of the angle

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00:24:37.998 --> 00:24:39.315
between the surface normal and the light direction

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00:24:39.315 --> 00:24:42.541
This simple formula allows shading to be 
efficiently calculated using the dot product

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00:24:42.542 --> 00:24:44.541
To apply this, first, calculate the surface normal vector

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00:24:44.541 --> 00:24:47.591
and the vector pointing from the surface to the light source

391
00:24:47.940 --> 00:24:53.461
Normalize the light vector so that its magnitude becomes 1,
then compute the dot product between the two vector

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00:24:53.461 --> 00:24:57.689
The result determines how much light is hitting the surface

393
00:24:57.829 --> 00:25:01.469
and serves as a key factor in defining the shading

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00:25:02.349 --> 00:25:05.701
The surface normal vector is typically denoted as N

395
00:25:05.701 --> 00:25:10.328
and the vector pointing from the surface to the light source is labeled as L

396
00:25:10.328 --> 00:25:12.581
It's commonly paired as N.L

397
00:25:12.870 --> 00:25:17.020
This basic formula for shading is

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00:25:17.021 --> 00:25:20.996
derived from Lambert’s Cosine Law

399
00:25:20.997 --> 00:25:23.077
and is the foundation for what is commonly called a diffuse shader

400
00:25:23.077 --> 00:25:27.621
For example, in game engines like Unity

401
00:25:27.622 --> 00:25:30.728
a diffuse shader utilizes this principle

402
00:25:30.728 --> 00:25:33.394
to define surface illumination

403
00:25:33.734 --> 00:25:39.516
based on the angle between the light

404
00:25:39.516 --> 00:25:40.821
That's the gist of this

405
00:25:40.821 --> 00:25:45.993
The N.L term represents

406
00:25:46.353 --> 00:25:48.463
the most fundamental lighting model

407
00:25:48.463 --> 00:25:51.100
and serves as the building block

408
00:25:51.101 --> 00:25:56.518
for more complex lighting calculations

409
00:25:57.359 --> 00:26:01.920
All you need to understand for now

410
00:26:01.920 --> 00:26:04.182
is that this N.L is at the fundamental level

411
00:26:04.920 --> 00:26:08.301
Lastly, I’ll explain the projection vector formula

412
00:26:08.501 --> 00:26:11.181
This formula is used

413
00:26:11.182 --> 00:26:12.562
to determine the projection

414
00:26:12.562 --> 00:26:14.679
of one vector onto another

415
00:26:14.679 --> 00:26:17.424
This formula comes handy in getting that

416
00:26:17.844 --> 00:26:20.285
When you have two vectors, u and v

417
00:26:20.960 --> 00:26:24.279
and you project u onto v, the resulting vector

418
00:26:24.496 --> 00:26:26.875
is what I’ll call v'

419
00:26:26.875 --> 00:26:32.064
This v' is the projection onto v

420
00:26:32.384 --> 00:26:34.486
So it is aligned in the same direction as v

421
00:26:34.486 --> 00:26:35.663
but with a different magnitude

422
00:26:36.175 --> 00:26:38.355
Different magnitude, same direction

423
00:26:38.923 --> 00:26:42.933
To calculate v’

424
00:26:42.934 --> 00:26:45.325
since the direction is the same and the magnitude different

425
00:26:45.325 --> 00:26:49.155
we first need to normalize v

426
00:26:49.155 --> 00:26:52.315
By normalizing it, we get this v-hat

427
00:26:52.315 --> 00:26:55.291
Create a unit vector v-hat with a magnitude of 1

428
00:26:55.303 --> 00:27:01.105
Then, by multiplying v-hat by the magnitude of v’

429
00:27:01.105 --> 00:27:03.971
we can determine v’

430
00:27:04.904 --> 00:27:09.225
But how do we calculate the magnitude of v’?

431
00:27:09.485 --> 00:27:12.316
How do we get its norm?

432
00:27:12.864 --> 00:27:14.205
That's the question here

433
00:27:14.205 --> 00:27:17.185
Since we know what u is

434
00:27:17.730 --> 00:27:26.089
he magnitude of v’ can be found by multiplying the magnitude of u

435
00:27:26.089 --> 00:27:28.703
by the cosine of the angle, cos(theta), between u and v

436
00:27:29.123 --> 00:27:35.963
So, how do we calculate cos(theta)?

437
00:27:36.163 --> 00:27:41.463
Cos(theta) can be determined using the dot product

438
00:27:41.823 --> 00:27:45.025
Revisiting the dot product equation

439
00:27:45.026 --> 00:27:50.128
u dot v is equal to the magnitude of u times

440
00:27:50.129 --> 00:27:51.642
the magnitude of v times cosine of theta

441
00:27:51.998 --> 00:27:54.066
The cosine of the angle between them, cos theta

442
00:27:54.066 --> 00:27:57.465
can ultimately be rewritten and calculated

443
00:27:57.466 --> 00:27:58.714
using this formula

444
00:27:59.264 --> 00:28:03.405
Now, if we substitute this value back into the equation

445
00:28:05.721 --> 00:28:06.721
and multiply them together

446
00:28:07.684 --> 00:28:10.465
we can derive the final formula for the projection vector

447
00:28:10.465 --> 00:28:11.415
This v'

448
00:28:11.706 --> 00:28:13.826
The final projection vector is now here

449
00:28:14.364 --> 00:28:19.025
Here, for the dot product u dot v

450
00:28:19.495 --> 00:28:23.194
the denominator consists of the magnitude of u by that of v

451
00:28:24.444 --> 00:28:28.125
For v-hat, the denominator is the magnitude of v

452
00:28:28.125 --> 00:28:29.537
and the numerator is v

453
00:28:29.751 --> 00:28:34.290
Since the magnitude of v is used twice

454
00:28:34.291 --> 00:28:36.503
the result becomes the square of the magnitude of v

455
00:28:36.503 --> 00:28:39.224
Therefore, the expression can be written as u.v.v

456
00:28:39.651 --> 00:28:42.290
Since the numerator and denominator involve u

457
00:28:42.870 --> 00:28:44.910
 any overlapping terms will cancel out

458
00:28:44.910 --> 00:28:46.644
The final formula for the projection vector is as follows

459
00:28:46.975 --> 00:28:48.935
The square of the magnitude

460
00:28:48.936 --> 00:28:51.893
is equivalent to the result of multiplying the same vector by itself twice

461
00:28:51.893 --> 00:28:53.864
so it can also be expressed as this

462
00:28:53.864 --> 00:28:56.069
Both are considered identical expressions

463
00:28:56.604 --> 00:29:00.961
If the magnitude of the vector v being projected onto is 1

464
00:29:00.962 --> 00:29:03.122
there is no need to calculate the magnitude

465
00:29:03.397 --> 00:29:06.677
and the formula simplifies further

466
00:29:07.203 --> 00:29:11.505
Now, this projection vector formula

467
00:29:11.835 --> 00:29:16.515
is significantly utilized in various vector applications

468
00:29:16.515 --> 00:29:20.903
and theoretical aspects, such as equations of planes

469
00:29:20.903 --> 00:29:22.903
and other mathematical theories

470
00:29:23.503 --> 00:29:29.225
For now, I’ll explain that this basic formula exists

471
00:29:29.226 --> 00:29:33.486
and I’ll provide a more detailed explanation of its applications

472
00:29:33.486 --> 00:29:35.856
later when we discuss plane equations

473
00:29:36.518 --> 00:29:39.125
This was on dot product

474
00:29:39.225 --> 00:29:41.305
This concludes today's lecture

475
00:29:41.305 --> 00:29:43.152
Thank you for listening

476
00:29:43.153 --> 00:29:44.113
Thank you

477
00:29:44.532 --> 00:29:45.832
Concept of Vector Dot Product
Vector Dot Product
Multiplying corresponding components and then summing them

478
00:29:45.833 --> 00:29:46.833
Properties of the dot product: Satisfies the commutative property, Not satisfy the associative property, Satisfies the distributive property
(a,b)x(c,d)=ac+bd

479
00:29:46.834 --> 00:29:47.834
The dot product of the same n-dimensional vector results in the square of its norm

480
00:29:47.835 --> 00:29:49.503
Cosine Formula of the Dot Product
axb=vector a x vector b x cosine theta
When the magnitudes of two vectors are 1, scalar a x scalar b = cosine theta

481
00:29:49.503 --> 00:29:50.983
Concept of vector inner product
Determining orthogonality of vector inner product
You can prove that two vectors are orthogonal by the inner product

482
00:29:50.983 --> 00:29:51.845
 (cos(theta), sin(theta)) dot (-sin(theta), cos(theta)) equals to 0

483
00:29:51.845 --> 00:29:52.985
Rigid transformation
A transformation that maintains the shape of an object
Rotation transformation does not change the shape of an object

484
00:29:52.985 --> 00:29:54.485


485
00:29:54.485 --> 00:29:55.426
Applications of the Dot Product
Front-Back Determination
If the dot product of two vectors is positive, the object is in front

486
00:29:55.427 --> 00:29:56.647
If the dot product is negative, the object is behind the viewing direction

487
00:29:56.647 --> 00:29:57.668
Field of View Check
The inner product of the line of sight vector and the vector to the target is cosine alpha
If the value of cosine alpha is greater than beta over cosine 2, the target is in the field of view

488
00:29:57.668 --> 00:29:58.589
Shading Calculation
Lambert’s Cosine Law: The brightness of light reflected is proportional to cos angle between surface and light direction

489
00:29:58.589 --> 00:29:59.451
Shader Lighting Equation N.L
The normal vector of the surface in space (N)
The vector pointing from the surface to the light source (L)

490
00:29:59.451 --> 00:30:02.522
Projection Vector Formula
Finding a vector projected onto another vector

491
00:30:02.522 --> 00:30:04.263
When the magnitude of the vector being projected is 1,
vector v’ is u dot v multiplied by v