If θ is the angle between two vectors ^i - 2^j + 3^k and 3^i - 2^j + ^k, find θ.

Given vectors are a = ^i - 2^j + 3^k and b = 3^i - 2^j + ^k,
If θ is the angle between the given vectors, then
cos θ = a.b / |a||b|
Now, a.b = (^i - 2^j + 3^k) . (3^i - 2^j + ^k) = 3 + 4 + 3 = 10
Also, |a| = √1² + (-2)² + 3² = √1 + 4 + 9 = √14
and |b| = √3² + (-2)² + 1² = √9 + 4 + 1 = √14
∴ cos θ = 10 / 14 = 5 / 7
⇒ θ = cos⁻¹(5/7)
Hence, θ = cos⁻¹(5/7) is the angle between the vectors a and b.