Show that the points whose position vectors are `5 hat i+5 hat k ,\ 2 hat i+ hat j+3 hat k\ a n d-4 hat i+3 hat j- hat k` are collinear.

Explanation

To show that the points whose position vectors are \( \vec{A} = 5\hat{i} + 5\hat{k}, \vec{B} = 2\hat{i} + \hat{j} + 3\hat{k}, \text{ and } \vec{C} = -4\hat{i} + 3\hat{j} - \hat{k} \) are collinear, we can use the determinant method. The points are collinear if the volume of the parallelepiped formed by the vectors is zero, which corresponds to the determinant of the matrix formed by these vectors being zero.

### Step 1: Write the position vectors in coordinate form
The position vectors can be expressed in coordinate form as:
- \( A(5, 0, 5) \)
- \( B(2, 1, 3) \)
- \( C(-4, 3, -1) \)

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