Prove that the points having position vectors ` hat i+2 hat j+3 hat k ,\ 3 hat i+4 hat j+7 hat k ,\ -3 hat i-2 hat j-5 hat k` are collinear.

Explanation

To prove that the points with position vectors \( \mathbf{a} = \hat{i} + 2\hat{j} + 3\hat{k} \), \( \mathbf{b} = 3\hat{i} + 4\hat{j} + 7\hat{k} \), and \( \mathbf{c} = -3\hat{i} - 2\hat{j} - 5\hat{k} \) are collinear, we can follow these steps:

### Step 1: Define the Position Vectors
Let:
- Point \( P \) have position vector \( \mathbf{a} = \hat{i} + 2\hat{j} + 3\hat{k} \)
- Point \( Q \) have position vector \( \mathbf{b} = 3\hat{i} + 4\hat{j} + 7\hat{k} \)
- Point \( R \) have position vector \( \mathbf{c} = -3\hat{i} - 2\hat{j} - 5\hat{k} \)

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