How to solve these computations

Question

Hey I'm looking to check my answers to make sure these computations are correct. The definition used is

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

Calculate u • v

Calculate u x v

Calculate (u x v)•w

Find the angle between u and v

All letters have the -> symbol above them.

If you could provide work showing how you did it too thatd be great.

Answer 1

Just going to write out the vectors in bracket notation:

u = < 1, 2, - 1>, v = <3, -1 , 4> w = <0, 2, -1>

a) Now we get to the dot product, which is just multiplying corresponding terms of the two vectors and adding them up:

u ⋅ v = 1*3 + 2*-1 + -1*4 = 3 - 2 - 4 = -3

b) Cross product is trickier to explain, so I just did the calculation:

u x v = <1 , 2, -1> x <3, -1 , 4> = det [ (i j k), (1 2 -1), (3 1 -4)] = (2*4 - -1*-1) i + (-1*3 - 1*4) j + (1*-1 - 2*-1)k

= <2*4 - -1*-1, -1*3 - 1*4, 1*-1 - 2*-1> = <8-1, -3 - 4, -1+2> = <7, -7, 1> = 7i - 7k + j

c) We know what u x v is from the previous part, so we can calculate (u x v) ⋅ w pretty easily

(u x v) ⋅ w = <7, -7, 1> ⋅<0, 2, -1> = 7*0 + -7*2 + 1*-1 = 0 - 14 - 1 = -15

d) Angle between u and v:

cos a = u ⋅ v / |u| * | v| = -3 / √(1² + 2²+ (-1)² * √3² + 1² + 4² = -3 / √1+4+1 * √9+1+16 = -3/√6 *26

= -3 / √156 = -3/2√39

a = arccos (-3 / 2√39), which means a ≈ 1.8134 radians