We can use the dot product formula to find the angle between the vectors:
a · b = ||a|| ||b|| cos(θ)
where ||a|| and ||b|| are the magnitudes of the vectors a and b, respectively.
First, we need to find the magnitudes of the vectors:
||a|| = sqrt(3^2 + 0^2 + (-1)^2) = sqrt(10)
||b|| = sqrt(2^2 + 0^2 + 2^2) = 2sqrt(2)
Then, we can calculate the dot product:
a · b = (3i - k) · (2i + 2k)
a · b = 6 + 0 - 2
a · b = 4
Now we can solve for the angle θ:
a · b = ||a|| ||b|| cos(θ)
4 = sqrt(10) * 2sqrt(2) cos(θ)
cos(θ) = 4 / (sqrt(10) * 2sqrt(2))
cos(θ) = 2 / sqrt(5)
Finally, we can calculate the angle θ:
θ = cos^-1(2 / sqrt(5))
θ ≈ 54.7 degrees
Therefore, the angle between a and b is approximately 54.7 degrees.