To find the angle θ between the two vectors u and v, we can use the formula:
cos(θ) = (u • v) / (|u| |v|)
First, we need to calculate the dot product of the vectors u and v:
u • v = (2 * 3) + (4 * 8) = 6 + 32 = 38
Next, we calculate the magnitudes of the vectors u and v:
|u| = √(2² + 4²) = √(4 + 16) = √20 = 2√5 ≈ 4.47
|v| = √(3² + 8²) = √(9 + 64) = √73 ≈ 8.54
Now we can substitute these values into the formula:
cos(θ) = 38 / (4.47 * 8.54) ≈ 38 / 38.2 ≈ 0.9947
Finally, we find the angle θ by taking the arccos of 0.9947:
θ ≈ arccos(0.9947) ≈ 8.1°
So, the angle between the given vectors u (2, 4) and v (3, 8) is approximately 8.1 degrees.