Given the vectors C = 2i + 7j and D = 14i + 38j, what is the angle between them?

To find the angle between the two vectors C and D, we can use the formula for the cosine of the angle θ between two vectors:

[

\cos(\theta) = \frac{{C \cdot D}}{{|C| |D|}}

]

First, let's calculate the dot product (C \cdot D):

[

C \cdot D = (2i + 7j) \cdot (14i + 38j) = (2 \times 14) + (7 \times 38) = 28 + 266 = 294

]

Next, we calculate the magnitudes of each vector:

[

|C| = \sqrt{2^2 + 7^2} = \sqrt{4 + 49} = \sqrt{53}

]

[

|D| = \sqrt{14^2 + 38^2} = \sqrt{196 + 1444} = \sqrt{1640}

]

Now, we can compute the cosine of the angle:

[

|D| = \sqrt{1640} \approx 40.5

]

This implies:

[

\cos(\theta) =