suppose u = i +2j-k, v=3i-j+4k and w=2j-k are three vectors in space calculate?

Question

1. u x v
2. u dot(u x v)
3.the projection of u x v along w
4. the angle between u x v and u

Answer 1

i=2
j=8
k=3
v=6
w=9

this balances the equation and help you figure the answer, i got 56

Answer 2

Given vectors:

u = i + 2j - k
v = 3i - j + 4k
w = 2j - k

calculate:

1) u X v = <1, 2, -1> X <3, -1, 4> = <7, -7, -7>

2) u • (u X v) = <1, 2, -1> • <7, -7, -7> = 7 - 14 + 7 = 0

3) the projection of u X v along w

First calculate the dot product of the two vectors.
(u X v) • w = <7, -7, -7> • <0, 2, -1> = 0 - 14 + 7 = -7

Calculate the magnitude of w.
|| w || = √[0² + 2² + (-1)²] = √(0 + 4 + 1) = √5

The scalar projection of (u X v) onto w is:
[(u X v) • w] / || w || = -7/√5

The vector projection of (u X v) onto w is:
{[(u X v) • w] / || w ||} * (w/|| w ||)
= {[(u X v) • w] / || w ||²} * w = (-7/5)<0, 2, -1>
= <0, -14/5, 7/5>

4) the angle between (u X v) and u

The cross product of two vectors is always perpendicular to both of them. So (u X v) is perpendicular to u. Therefore the angle between (u X v) and u is 90°. Indeed from 2) we see that the dot product of the two vectors is zero as would be exptected from perpendicular vectors.

Answer 3

u.v= 1*3 + 2*-1 +-1*-1= 3-2+1= 2

Answer 4

u cross v:
-k- 4j- 6k+ 8i-3j+i = 9i - 7j - 7k

u dot above:
9-14+7 = 2

I don't know the other two...