Prove that if `cos alpha ne 1, cos beta ne1 and cos gamma ne 1`, then the vectors `a=hati cos alpha+hatj+hatk,b=hati+hatj cos beta+hatk` and `c=hati+hatj+hatk cos gamma` can never be coplanar.

Explanation

To prove that the vectors \( \mathbf{a} = \hat{i} \cos \alpha + \hat{j} + \hat{k} \), \( \mathbf{b} = \hat{i} + \hat{j} \cos \beta + \hat{k} \), and \( \mathbf{c} = \hat{i} + \hat{j} + \hat{k} \cos \gamma \) are non-coplanar when \( \cos \alpha \neq 1 \), \( \cos \beta \neq 1 \), and \( \cos \gamma \neq 1 \), we can use the determinant method.

### Step 1: Write the vectors in matrix form
We can express the vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) in a matrix form to find the determinant:

\[
\begin{vmatrix}
\cos \alpha & 1 & 1 \\
...