Calculus 3 - Unit 12

Points in 3D space, vectors

vector projᵥu or proj u(v)

Algebraic: (vu)/(|v|)*(v/|v|) (length of |p| * scale to length 1)

Geometric: (-)|u|cosθ(v/|v|)

Negative if θ is obtuse. Zero if θ is 90.

Power-reducing formula

(cos2θ+1)/2 = cos²θ

(1-cos2θ)/2 = sin²θ

Standard form for circles centered at (c₁,c₂,c₃)

(x-c₁)² + (x-c₂)² + (x-c₃)² = r²

Magnitude/length of a vector

|v| = √(v₁² + v₂²)

Unit vector

Vector with magnitude = 1

|u| = (1/|v|) * (|v|) = 1

Finding vector in direction <given> with length #.

1. Treat <given> like a unit vector. Find its magnitude.

2. Multiply <given> vector by 1/magnitude found.

3. Multiply the vector by the length given.

Dot product

Algebraic: u * v = u₁v₁ + u₂v₂

Geometric: u v = |u||v|*cosθ (can be used to find θ between two vectors)

Multiplying unit vectors i = <1,0> and j = <0,1>

Same x same, = 1 (parallel)

Different x different = 0 (perpendicular)

Rewriting a = <3, 2, 5> using standard unit vectors

a = 3i + 2j + 5k

Find the unit vectors parallel to the tangent line f(x) through (given coordinates)

1. Find the tangent line of f(x) at the given coordinates.

2. Find an ordered pair in the tangent line that represents tangent line slope (ex: slope = 3, use point (1, 3)

3. Treat ordered pair as a vector, and divide it by its magnitude to get a unit vector.

Is the angle between a and b acute, obtuse, or 90?

If a*b > 0 , acute.

If a*b < 0 , obtuse.

If a*b = 0 , a and b are perpendicular, and θ = π/2

If a and b point in the same direction, θ = 0, and a*b = |a||b|

If a and b point in opposite directions, θ = π and a*b = -|a||b|

Standard base vectors/standard unit vectors

i = <1,0,0> j = <0,1,0> k = <0,0,1>

Determinate of 2×2 matrix

ad - bc

Determinate of 3×3 matrix

Cross-product

Properties of cross-products

The vector a x b is perpendicular to both a and b

Two nonzero vectors are parallel if |a x b| = 0

|a x b| = |a||b|sinθ

Volume of parallelepided determed by vectors a, b, and c.

Magnitude of their scalar triple product

V = |a * (b x c)|

Scalar projection or compᵥu

(vu)/(|v|)

Orthogonal projection of u onto v or orthᵥu

u - projᵥu

Angle between two nonzero vectors u and v

cosθ = (ab)/|a||b|

Find a decomposition of a = <vector> into a vector c parallel to b <vector> and a vector d perpendicular to b such that c + d = a

1. Let c = proj_ba

2. Let d = a - c

Area of parallelogram made with u and v

|u x v|

Area of triangle is ½ of this, using any two adjacent vectors

(ex: given points A, B, and C, use vectors AB and AC.)