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

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

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

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

To find the angle between two vectors, u and w, you can use the dot product formula:

θ = cos^(-1)((u · w) / (|u| * |w|))

where u · w represents the dot product of vectors u and w, and |u| and |w| represent the magnitudes of vectors u and w, respectively.

a) Let's find the angle between vectors u and w:

1. Find the dot product of u and w:
u · w = (-2 * 4) + (-1 * 3) + (2 * -2)

2. Find the magnitudes of u and w:
|u| = sqrt((-2)^2 + (-1)^2 + 2^2)
|w| = sqrt(4^2 + 3^2 + (-2)^2)

3. Calculate the angle θ:
θ = cos^(-1)((u · w) / (|u| * |w|))

Now let's proceed to find out if vectors u, v, and w are coplanar or not.

b) For three vectors to be coplanar, their scalar triple product must be equal to zero.

The scalar triple product is given by the formula:

u ⨂ (v ⨂ w) = u · (v x w)

where u · (v x w) represents the dot product of vector u with the cross product of vectors v and w.

1. Calculate the cross product of vectors v and w:
v x w = (v2 * w3 - v3 * w2, v3 * w1 - v1 * w3, v1 * w2 - v2 * w1)

2. Calculate the dot product of u with the cross product of v and w:
u · (v x w) = u1 * (v2 * w3 - v3 * w2) + u2 * (v3 * w1 - v1 * w3) + u3 * (v1 * w2 - v2 * w1)

If the result of the dot product is equal to zero, it means that vectors u, v, and w are coplanar.