What is the angle in radians between the vectors a = < -3,3,3 > and b = <-4,1,-5 > ?

Answer 1

One of the ways to define the dot product between two vectors is:

a ⋅ b = |a| |b| cos θ

where |a| represents the magnitude (or modulus) of the vector a

and θ is the angle between the two vectors.

Here:

a ⋅ b = <-3, 3, 3> ⋅ <-4, 1, -5> = -3 * (-4) + 3 * 1 + 3 * (-5) = 12 + 3 - 15 = 15 - 15 = 0

So we do not even need to find the magnitudes of the vectors.

cos θ = (a ⋅ b) / (|a| |b|) = 0

Thus, the angle between the two vectors is θ = π / 2.

The two vectors are orthogonal.