Find the value of `lambda` for which the four points with position vectors ` hat j- hat k ,\ 4 hat i+5 hat j+lambda hat k ,\ 3 hat i+9 hat j+4 hat k\ a n d-4 hat i+4 hat j+4 hat k` are coplanar.

Explanation

To find the value of \( \lambda \) for which the four points with position vectors are coplanar, we can use the scalar triple product. The four points are given by their position vectors:

1. \( \mathbf{A} = -\hat{j} - \hat{k} \)
2. \( \mathbf{B} = 4\hat{i} + 5\hat{j} + \lambda\hat{k} \)
3. \( \mathbf{C} = 3\hat{i} + 9\hat{j} + 4\hat{k} \)
4. \( \mathbf{D} = -4\hat{i} + 4\hat{j} + 4\hat{k} \)

### Step 1: Find the vectors \( \mathbf{AB} \), \( \mathbf{AC} \), and \( \mathbf{AD} \)
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