vectors jeff chastine 1. a mathematical structure that has more than one “part” (e.g. an array)...

JEFF CHASTINE 1

VECTORS

JEFF CHASTINE 2

�⃗�𝑒𝑐𝑡𝑜𝑟𝑠• A mathematical structure that has more than one “part” (e.g. an array)

• 2D vectors might have x and y

• 3D vectors might have x, y and z

• 4D vectors might have x, y, z and w

• Vectors can represent a point in space

• Vectors commonly represent both:

• Direction

• Magnitude

JEFF CHASTINE 3

�⃗�𝑒𝑐𝑡𝑜𝑟𝑠• A vector is often denoted with an arrow above it (e.g.)

• Row representation [x, y, z] (multiple columns)

• Column vector has multiple rows

• Vectors will be used in lighting equations

[ 15−43 ]

JEFF CHASTINE 4

EXAMPLE• How would I describe the 2D difference in location of a player and an enemy?

You

Man-Bear-Pig

JEFF CHASTINE 5

EXAMPLE• How would I describe the 2D difference in location of a player and an enemy?

(x1, y1)

(x2, y2)

6

EXAMPLE• How would I describe the 2D difference in location of a player and an enemy?

(x1, y1)

(x2, y2)

Note: a very useful 2D function is atan2 (y, x) which gives you the angle!

…or

�⃗�𝑖𝑓𝑓 =[(𝑥1−𝑥2 ) , ( 𝑦1− 𝑦2 )]

JEFF CHASTINE 7

INTERPRETATION• has both magnitude and direction

(x1, y1)

(x2, y2)

Magnitude (length)

JEFF CHASTINE 8

INTERPRETATION• has both magnitude and direction

(x1, y1)

(x2, y2)

Direction

(∆x, ∆y)

(0, 0)

JEFF CHASTINE 9

ADDING/SUBTRACTING VECTORS• Do this component-wise

• Therefore, the vectors must be the same size

• Adding example

• + =

• Subtraction works the same way

+ =

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MULTIPLICATION?• Multiplying by a scalar (a single number)

• 6 * =

• What about multiplication?

• This isn’t really defined, but we do have

• Dot product

• Cross product

JEFF CHASTINE 11

MAGNITUDE AND NORMALIZATION OF VECTORS• Normalization is a fancy term for saying the vector should be of length 1

• Magnitude is just its length and denoted

• Example for

• mag =

• mag =

• To normalize the vector, divide each component by its magnitude

• Example from above

• Magnitude of

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THE DOT PRODUCT• Also called the inner or scalar product

• Multiply component-wise, then sum together

• Denoted using the dot operator

• Example

• Why is this so cool?

• If normalized, it’s the cosine of the angle θ between the two vectors!

• Use to “undo” that

• Basis of almost all lighting calculations!

𝜃

JEFF CHASTINE 13

DOT PRODUCT EXAMPLE• Assume we have two vectors:

• These vectors are already normalized

• We expect the angle to be 90°

• Dot product is:

𝜃�⃗�

𝑣

JEFF CHASTINE 14

DOT PRODUCT EXAMPLE 2• Assume we have two vectors:

• We expect the angle to be 180°

• Dot product is:

𝜃

�⃗�𝑣

JEFF CHASTINE 15

DOT PRODUCT EXAMPLE 3• Assume we have two vectors:

• We expect the angle to be 0°

• Dot product is: �⃗⃗�𝑣

JEFF CHASTINE 16

PROJECTION• Can also be used to calculate the projection of one vector onto another

‖𝑉‖cos (𝛼 )= 𝑉 ∙𝑊‖𝑉‖∙‖𝑊‖

𝑉

𝑊𝛼

𝑝𝑟𝑜𝑗𝑤𝑉=𝑉 ∙𝑊

‖𝑊‖2𝑊

Length of projection is: Then, multiply by normalized

JEFF CHASTINE 17

CROSS PRODUCT• Gives us a new vector that is perpendicular to the other two

• Denoted with the × operator

• Calculations:

• Interesting:

• If then

• The magnitude (length) of the new vector is the sine of the angle (if normalized)

𝜃

�⃗�

�⃗�

�⃗�

|𝑎𝑥 𝑎𝑦 𝑎𝑧𝑏𝑥 𝑏𝑦 𝑏 𝑧|

JEFF CHASTINE 18

CROSS PRODUCT• Gives us a new vector that is perpendicular to the other two

• Denoted with the × operator

• Calculations:

• Interesting:

• If then

• The magnitude (length) of the new vector is the sine of the angle (if normalized)

𝜃

�⃗�

�⃗�

�⃗�

|𝑎𝑥 𝑎𝑦 𝑎𝑧𝑏𝑥 𝑏𝑦 𝑏 𝑧|

JEFF CHASTINE 19

CROSS PRODUCT• Gives us a new vector that is perpendicular to the other two

• Denoted with the × operator

• Calculations:

• Interesting:

• If then

• The magnitude (length) of the new vector is the sine of the angle (if normalized)

𝜃

�⃗�

�⃗�

�⃗�

|𝑎𝑥 𝑎𝑦 𝑎𝑧𝑏𝑥 𝑏𝑦 𝑏 𝑧|

JEFF CHASTINE 20

CROSS PRODUCT• Gives us a new vector that is perpendicular to the other two

• Denoted with the × operator

• Calculations:

• Interesting:

• If then

• The magnitude (length) of the new vector is the sine of the angle (if normalized)

𝜃

�⃗�

�⃗�

�⃗�

|𝑎𝑥 𝑎𝑦 𝑎𝑧𝑏𝑥 𝑏𝑦 𝑏 𝑧|

JEFF CHASTINE 21

FUN QUESTIONS• How do we find the normal of a triangle?

• How can we determine if a polygon is facing away from the camera?

JEFF CHASTINE 22

FUN QUESTIONS• How do we find the normal of a triangle?

• How can we determine if a polygon is facing away from the camera?

P0

P1

P2

JEFF CHASTINE 23

FUN QUESTIONS• How do we find the normal of a triangle?

• How can we determine if a polygon is facing away from the camera?

P2

P0

P1

�⃗� 𝑣

Make some vectors…

JEFF CHASTINE 24

FUN QUESTIONS• How do we find the normal of a triangle?

• How can we determine if a polygon is facing away from the camera?

P2

P0

P1

�⃗�𝑣

Make some vectors…

JEFF CHASTINE 25

FUN QUESTIONS• How do we find the normal of a triangle?

• How can we determine if a polygon is facing away from the camera?

P2

P0

P1

�⃗�𝑣

Take the cross product

�⃗�×𝑣

JEFF CHASTINE 26

FUN QUESTIONS• How do we find the normal of a triangle?

• How can we determine if a polygon is facing away from the camera?

P2

P0

P1

�⃗�

JEFF CHASTINE 27

FUN QUESTIONS• How do we find the normal of a triangle?

• How can we determine if a polygon is facing away from the camera?

P2

P0

P1

�⃗�

JEFF CHASTINE 28

FUN QUESTIONS• How do we find the normal of a triangle?

• How can we determine if a polygon is facing away from the camera?

P2

P0

P1camera

�⃗�

JEFF CHASTINE 29

FUN QUESTIONS• How do we find the normal of a triangle?

• How can we determine if a polygon is facing away from the camera?

P2

P0

P1camera�⃗�𝑎𝑚

�⃗�

JEFF CHASTINE 30

FUN QUESTIONS• How do we find the normal of a triangle?

• How can we determine if a polygon is facing away from the camera?

P2

P0

P1camera�⃗�𝑎𝑚

�⃗�

𝐼𝑓 acos (𝑁 ∙𝑐𝑎𝑚 )<90 ° ,𝑖 𝑡 ′ 𝑠𝑣𝑖𝑠𝑖𝑏𝑙𝑒

Assuming and are normalized

JEFF CHASTINE 31

A FINAL NOTE• Can multiply a matrix and vector to:

• Rotate the vector

• Translate the vector

• Scale the vector

• Etc..

• This operation returns a new vector

JEFF CHASTINE 32

THE END

Image of a triangle facing away from the camera