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` and `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 will follow these steps:

### Step 1: Define the position vectors
Let the position vectors of the points be defined as follows:
- \( \mathbf{A} = -\hat{j} - \hat{k} \)
- \( \mathbf{B} = 4\hat{i} + 5\hat{j} + \lambda\hat{k} \)
- \( \mathbf{C} = 3\hat{i} + 9\hat{j} + 4\hat{k} \)
- \( \mathbf{D} = 4\hat{i} + 4\hat{j} + 4\hat{k} \)
...