The angle between the vectors v and w can be found using the formula:
cos(θ) = (v • w) / (||v|| ||w||)
Where v • w is the dot product of v and w, and ||v|| and ||w|| are the magnitudes of v and w, respectively.
First, let's calculate the dot product of v and w:
v • w = (2 * 6) + (1 * 3) + (3 * 9)
= 12 + 3 + 27
= 42
Next, let's calculate the magnitudes of v and w:
||v|| = √(2^2 + 1^2 + 3^2)
= √(4 + 1 + 9)
= √14
||w|| = √(6^2 + 3^2 + 9^2)
= √(36 + 9 + 81)
= √126
Now, we can substitute these values into the formula:
cos(θ) = 42 / (√14 * √126)
= 42 / (√(14 * 126))
= 42 / (√(1764))
= 42 / 42
= 1
Finally, we can solve for θ:
θ = cos^(-1)(1)
= 0°
So, the angle between the vectors v and w is 0°.