Tristin S. answered • 02/22/21

Recent College Graduate Looking for Opportunities to Tutor Others

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 / √(1^{2} + 2^{2}+ (-1)^{2} * √3^{2} + 1^{2} + 4^{2} = -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