Find the angle between the force `vecF(5hati+4hatj-3hatk)`units and displacement `vecd=(3hati+4hatj+5hatk)` unit. Also find the projection of `vecF` on `vecd`

Explanation

To find the angle between the force vector \(\vec{F} = 5\hat{i} + 4\hat{j} - 3\hat{k}\) and the displacement vector \(\vec{d} = 3\hat{i} + 4\hat{j} + 5\hat{k}\), we will use the dot product formula. We will also find the projection of \(\vec{F}\) onto \(\vec{d}\).

### Step 1: Calculate the dot product \(\vec{F} \cdot \vec{d}\)

The dot product of two vectors \(\vec{A} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\) and \(\vec{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}\) is given by:

\[
\vec{A} \cdot \vec{B} = a_1b_1 + a_2b_2 + a_3b_3
...