The position vectors of the points `P ,\ Q ,\ R` are ` hat i+2 hat j+3 hat k ,\ -2 hat i+3 hat j+5 hat k\ a n d\ 7 hat i- hat k` respectively. Prove that `P ,\ Q\ a n d\ R` are collinear points.

Explanation

To prove that the points \( P, Q, R \) are collinear, we will use the concept of vectors. Specifically, we will show that the vectors \( \vec{PQ} \) and \( \vec{PR} \) are parallel. If two vectors are parallel, it implies that the points they connect are collinear.

### Step 1: Write down the position vectors of points \( P, Q, R \).

The position vectors are given as:
- \( \vec{P} = \hat{i} + 2\hat{j} + 3\hat{k} \)
- \( \vec{Q} = -2\hat{i} + 3\hat{j} + 5\hat{k} \)
- \( \vec{R} = 7\hat{i} - \hat{k} \)
...