True or False : The angle between the vectors < 6, 2, -4> and <2, 1, -3> is 106.6 degrees ?

Answer 1

False

the dot product = 26
and the product of their lengths are = 28
arccos(26/28) = 21.8 degrees

Answer 2

True or False : The angle between the vectors < 6, 2, -4> and <2, 1, -3> is 106.6 degrees.

The dot product of two vectors u and v is:

u • v = || u || || v || cosθ
where θ is the angle between the vectors.

cosθ = (u • v) / || u || || v ||

For the question at hand:

u = <6, 2, -4>
v = <2, 1, -3>

u • v = <6, 2, -4> • <2, 1, -3> = 6*2 + 2*1 - 4*(-3) = 26

|| u || = √[6² + 2² + (-4)²] = √(36 + 4 + 16) = √56 = 2√14
|| v || = √[2² + 1² + (-3)²] = √(4 + 1 + 9) = √14

cosθ = (u • v) / || u || || v || = 26 / [(2√14)*(√14)] = 26/28
cosθ = 13/14

θ = arccos(13/14) ≈ 21.8°

So the answer is false.