Find the value of `lambda` so that the following vectors are coplanar: ` vec a= hat i- hat j+ hat k , vec b=2 hat i+ hat j- hat k , vec c=lambda hat i- hat j+lambda hat k`

Explanation

To find the value of \( \lambda \) such that the vectors \( \vec{a}, \vec{b}, \vec{c} \) are coplanar, we can use the property that three vectors are coplanar if the scalar triple product of the vectors is zero. This can be represented using the determinant of a matrix formed by the vectors.

Given:
- \( \vec{a} = \hat{i} - \hat{j} + \hat{k} \)
- \( \vec{b} = 2\hat{i} + \hat{j} - \hat{k} \)
- \( \vec{c} = \lambda \hat{i} - \hat{j} + \lambda \hat{k} \)

### Step 1: Set up the determinant
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