Show that the vectors ` -> a= hat i-2\ hat j+3 hat k , -> b=\ 2 hat i+3j-4 hat k\ and\ c= hat i-3\ hat j+5 hat k`are coplanar.

Explanation

To show that the vectors \( \vec{a} = \hat{i} - 2\hat{j} + 3\hat{k} \), \( \vec{b} = 2\hat{i} + 3\hat{j} - 4\hat{k} \), and \( \vec{c} = \hat{i} - 3\hat{j} + 5\hat{k} \) are coplanar, we can use the scalar triple product. The vectors are coplanar if the scalar triple product \( \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \).

### Step 1: Find the cross product \( \vec{b} \times \vec{c} \)

The cross product of two vectors can be calculated using the determinant of a matrix formed by the unit vectors and the components of the vectors:

\[
\vec{b} \times \vec{c} = \begin{vmatrix}
...