` vec A B=3 hat i- hat j+ hat ka n d vec C D=-3 hat i+2 hat j+4 hat k` are two vectors. The position vectors of the points `Aa n dC` are `=6 hat i+7 hat j+4 hat ka n d=-9 hat j+2 hat k` respectively. Find the position vector of a point `P` on the line `A B` and a point `Q` on the line `C D` such that ` vec P Q` is perpendicular to ` vec A Ba n d vec C B` both.

Explanation

To solve the problem, we need to find the position vectors of points \( P \) and \( Q \) such that the vector \( \vec{PQ} \) is perpendicular to both \( \vec{AB} \) and \( \vec{CD} \). Let's break this down step by step.

### Step 1: Define the Vectors
Given:
- \( \vec{AB} = 3 \hat{i} - \hat{j} + \hat{k} \)
- \( \vec{CD} = -3 \hat{i} + 2 \hat{j} + 4 \hat{k} \)

Position vectors of points \( A \) and \( C \):
...