Vectors

Question 1

||v||

Answer 1

Norm of v = ((v1)^2 + … + (vn)^2)^1/2

Question 2

Dot product of v • w

Answer 2

v1w1 + v2w2 + … + vnwn

Question 3

How to find cosØ in R2 and R3

Answer 3

(v • w)/((||v||)(||w||))

Question 4

cross product of v x w

Answer 4

1. Method where you write the vectors below
2. (v2w3 - v3w2)
   (v3w1 - v1w3)
   (v1w2 - v2w1)

Question 5

Is the cross product associative

Answer 5

No, u x (v x w) ≠ (u x v) x w

Question 6

How to find the direction of the cross product

Answer 6

Right hand rule

Question 7

Triple product

Answer 7

u • (v x w) = w • (u x v) = v • (w x u), invariance under cyclic permeations

Question 8

u, v and w are coplanar if and only if…

Answer 8

u • (v x w) = 0

Question 9

Vector triple product

Answer 9

u x (v x w) = (u • w)v - (u • v)w

Question 10

Lines vector equation form

Answer 10

x = x0 + tv
t is real, line is parallel to v through x0

Question 11

Planes vector equation form

Answer 11

x = x0 + t1v1 + t2v2
t1,t2 are real

Question 12

Distance p from line L where L: x = x0 +tv

Answer 12

D = ||(p - x0) x v||/(||v||)

Question 13

Distance p from plane P where
P: n • (x - x0) = 0

Answer 13

D = |(p - x0) • n|/(||n||)

Question 14

Projection theory where
v = w1 + w2
w1 is orthogonal to w2
w1 is parallel to a

Answer 14

w1 = proja(v) = ((a • v)/(||a||^2))a

w2 = v - w1