Find the value of `p` for which the vectors ` vec a=3 hat i+2 hat j+9 hat k` and `vec b= hat i+p hat j+3 hat k` are (i) perpendicular (ii) parallel

Explanation

To find the value of \( p \) for which the vectors \( \vec{a} = 3 \hat{i} + 2 \hat{j} + 9 \hat{k} \) and \( \vec{b} = \hat{i} + p \hat{j} + 3 \hat{k} \) are (i) perpendicular and (ii) parallel, we will solve each case step by step.

### Part (i): Finding \( p \) when the vectors are perpendicular

1. **Understanding the condition for perpendicularity**:
Two vectors are perpendicular if their dot product is zero. Therefore, we need to calculate \( \vec{a} \cdot \vec{b} \) and set it equal to zero.

2. **Calculating the dot product**:
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