Find the angle between the vector `vec(a) =2 hat(i) + 3hat(j) - 4 hat(k) and vec(b) = 4hat(i) +5 hat(j) - 2hat(k)` .

Explanation

To find the angle between the vectors \(\vec{a} = 2 \hat{i} + 3 \hat{j} - 4 \hat{k}\) and \(\vec{b} = 4 \hat{i} + 5 \hat{j} - 2 \hat{k}\), we can use the dot product formula. Here are the steps to solve the problem:

### Step 1: Calculate the Dot Product of Vectors \(\vec{a}\) and \(\vec{b}\)

The dot product of two vectors \(\vec{a}\) and \(\vec{b}\) is given by:

\[
\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z
...