Given the following: vector u= [-2,-1,2] and vector v=[1,-1,-4] and vector w= [4,3,-2] find:

Question

a)theta, the angle between vectors, u and w.

b)Are the vectors, u, v, and w coplanar?Show your answer algebraically.

Answer 1

u dot w = |u| |w| cosØ
[-2,-1,2]dot[4,3,-2] = √9√29cosØ
(-8 -3 -4)/3√29 = cosØ
Ø = appr 158.2°

to be coplanar, w must be a linear combination of u and v, that is
w = au +bv
[4,3,-2] = a[-2,-1,2] + b[1,-1,-4]

so -2a + b = 4
and -a - b = 3
add them:
-3a = 7
a = -7/3, then b = -2/3

if they are complanar, these values must satisfy
2a - 4b = -2
LS = 2(-7/3) - 4(-2/3)
= -14/3 + 8/3
= -2
= RS

Yes, they are coplanar