Chapter 4,

Chapter 4, "Vector Operations"

Vector Versus Scalar

Scalar = magnitude only.

Vector = magnitude + direction.

Displacement

Displacement = final position “ initial position.

D x = x _f “ x _i

Polar Coordinates

Vector

where is the magnitude of A and q is the direction.

Cartesian Coordinates (Components)

Vector

where is one unit in the x direction and is one unit in the y direction.

Converting from Polar to Cartesian Coordinates

For vector ,

where a ₁ = cos q and a ₂ = sin q .

Converting from Cartesian to Polar Coordinates

For vector ,

.

Cartesian Coordinates (Components) in 3D

Vector

where is one unit in the x direction, is one unit in the y direction, and is one unit in the z direction.

Commutative Law of Vector Addition

A + B = B + A

for any vectors A and B.

Adding 2D Vectors Numerically

for vectors and .

Adding 3D Vectors Numerically

for vectors and .

Subtracting Vectors Numerically

for vectors and .

Subtracting 3D Vectors Numerically

for vectors and .

Scalar Multiplication in Polar Coordinates

for any scalar c and vector .

Scalar Multiplication in Cartesian Coordinates

for any scalar c and vector .

Normalizing a 2D Vector

for any vector A = [ a ₁ a ₂ ].

Normalizing a 3D Vector

for any vector A = [ a ₁ a ₂ a ₃ ].

Dot Product in 2D

A B = a ₁ b ₁ + a ₂ b ₂

for any 2D vectors A = [ a ₁ a ₂ ] and B = [ b ₁ b ₂ ].

Dot Product in 3D

A B = a ₁ b ₁ + a ₂ b ₂ + a ₃ b ₃

for any 3D vectors A = [ a ₁ a ₂ a ₃ ] and B = [ b ₁ b ₂ b ₃ ].

Perpendicular Check

If A B = 0, A B.

Positive or Negative Dot Product

If A B < 0 (negative), q > 90 °

If A B > 0 (positive), q < 90 °

where q is the angle between vectors A and B.

Angle Between Two Vectors

where q is the angle between vectors A and B.

Cross-Product

A x B = [( a ₂ b ₃ “ a ₃ b ₂ ) ( a ₃ b ₁ “ a ₁ b ₃ ) ( a ₁ b ₂ “ a ₂ b ₁ )]

for any two vectors A = [ a ₁ a ₂ a ₃ ] and B = [ b ₁ b ₂ b ₃ ].

Perpendicular Vectors

A x B is perpendicular to both vectors A and B.

Cross-Product Is Not Commutative

A x B B x A

In fact, A x B = “(B x A) for any two 3D vectors A and B.

Surface Normal

Surface normal =

for any two 3D vectors A and B.

Angle Between Two Vectors

for any two 3D vectors A and B.