/linear algebra

dot product:

[a,b,c] * [x,y,z] = (ab) + (by) + (c*z)

magnitude (length):

∥[a,b,c]∥ = sqrt(a^2 + b^2 + c^2)

distance:

∥u-v∥ = sqrt((u1-v2)^2 + (u2-v2) + (u3-v3)^2)

angle between:

cos^-1 (u*v)/(∥u∥∥v∥)

projection:

proj(u,v) = (uv)/(uu) * u

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Normal Form: n * (x - p) = 0

General Form: Ax + By + C = 0

Vector Form: [x, y] = P + td

Parametric Form: x = _ + _t, y = _ + _t

Equation of line passing through P with normal vector n

P = (0,0), n = [3,2]

• Normal Form:

n * (x - p) = 0 [3, 2] * ([x, y] - [0, 0]) = 0

• General Form:

Ax + By + C = 0 3x + 2y = 0

P = (1,2), n = [3,-4]

• Normal Form:

n * (x - p) = 0 [3,-4] * ([x,y] - [1,2]) = 0 [3x - 3, -4y + 8] = 0 [3x,-4y] = -5

Equation of line passing through P with direction vector d

P = (1,0), d = [-1,3]

• Vector form:

x = P + td [x,y] = [1,0] + t[-1,3]

• Parametric form:

x = 1 - t y = 3t

P = (3,0,-2), d = [2,5,0]

• Vector form:

x = P + td [x,y] [3,0,-2] + t[2,5,0]

• Parametric form:

x = 3 + 2t y = 5t z = -2

Equation of plane passing thorugh P with normal vector n

P = (0,1,0), n = [3,2,1]

• Normal form:

n * (x - p) = 0 [3,2,1] * ([x,y,z] - [0,1,0]) = 0

• Parametric form:

3x + 2y - 2 + z = 0 3x + 2y + z = 2

Equation of plane passing thorugh P with direction vectors u and v

P = (0,0,0), u = [2,1,2], v = [-3,2,1]

• Vector form:

x = p + su + tv [x,y,z] = [0,0,0] + s[2,1,2] + t[-3,2,1]

• Parametric form:

x = 2s - 3t y = s + 2t z = 2s + t

Vector equation of line passing thorugh P and Q

P = (1,-2), Q = (3,0)

• Vector form:

x = p + td d = direction vector = PQ = Q - P d = [3,0] - [1,-2] = [2,2]

[x,y] = [1,-2] + t[2,2]

Vector equation of plane passing through P, Q, and R

P = (1,1,1), Q = (4,0,2), R = (0,1,-1)

• Vector form:

[x,y,z] = p + su + tv

u = PQ u = Q - P = [4,0,2] - [1,1,1] u = [3,-1,1]

v = PR v = R - P = [0,1,-1] - [1,1,1] v = [-1,0,-2]

[x,y,z] = [1,1,1] + s[3,-1,1] + t[-1,0,-2]

Convert slope form to vector and parametric form

y = 3x - 1

• Parametric form

Set one var (x or y) to t x = t y = 3t - 1

• Vector form

[x,y] = P + td [x,y] = [0,1] + t[1,3]

3x + 2y = 5

• Parametric form

Set one var (x or y) to t x = t y = 5/2 - 3/2t

• Vector form

[x,y] = P + td [x,y] = [0,5/2] + t[1,-3/2]

Vector equation of line passing through P and parallel to line

P = (2,-1), Line = 2x - 3y = 1

• Vector equation

[x,y] = P + td

2x - 3y = 1 normal vector = [2,-3] direction vector = [3,2] (Change sign of first)

[x,y] = [2,-1] + t[3,2]

Vector equation of line that passes through P and parallel to line

P = (-1,0,3) x = 1 - t, y = 2 + 3t, z = -2 -t

x = 1 - t y = 2 + 3t z = -2 -t

• Vector form

[x,y,z] = [1,2,-2] + t[-1,3,-1] [x,y,z] = [-1,0,3] + t[-1,3,-1]

Determine if planes are parallel, perpendicular, or neither

P1 = 4x - y + 5z = 2, P2 = 2x + 3y - z = 1

Two vectors are perpendicular iff their dot product is 0

[4,-1,5] * [2,3,-1] = 8 - 3 - 5 = 0 Perpendicular

Two vectors are parallel iff their divided components are equal Not Parallel