Show that the four points A, B, C and D with position vectors`4 hat i+5\ hat j+ hat k ,-(\ hat j+ hat k),3 hat j+9\ hat j+4 hat k\ a n d\ -4 hat i+ 4hat j+ 4hat k`, respectively are coplanar.

Explanation

To show that the four points A, B, C, and D with given position vectors are coplanar, we can use the concept of vectors and determinants. The points are coplanar if the volume of the parallelepiped formed by the vectors is zero, which can be determined using the scalar triple product or the determinant of a matrix formed by the vectors.

### Step-by-Step Solution:

1. **Identify the Position Vectors**:
- Let the position vectors of points A, B, C, and D be:
- \( \vec{A} = 4\hat{i} + 5\hat{j} + \hat{k} \)
- \( \vec{B} = -\hat{j} - \hat{k} \)
...