The question is asking for the value of $p$ for which the vectors $\mathbf{a} = 3\hat{i} + p\hat{j} + 9\hat{k}$ and $\mathbf{b} = \hat{i} + \hat{j} + 3\hat{k}$ are collinear.

Explanation:

Two vectors are collinear if one is a scalar multiple of the other. In this case, the vector $\mathbf{a}$ must be a scalar multiple of $\mathbf{b}$. So, we can express this relationship as: $\mathbf{a} = \lambda \mathbf{b}$ where $\lambda$ is some scalar constant.

This implies: $3\hat{i} + p\hat{j} + 9\hat{k} = \lambda ( \hat{i} + \hat{j} + 3\hat{k} )$

Expanding the right-hand side: $\lambda \hat{i} + \lambda \hat{j} + 3\lambda \hat{k}$

Now, equate the components of the vectors on both sides:

1. From the $\hat{i}$-component: $3 = \lambda$

2. From the $\hat{j}$-component: $p = \lambda$

3. From the $\hat{k}$-component: $9 = 3\lambda$

Solving the third equation: $9 = 3\lambda \quad \Rightarrow \quad \lambda = 3$

Thus, from the $\hat{i}$-component and the $\hat{j}$-component, we get $\lambda = 3$ and $p = 3$.

Conclusion:

The value of $p$ is 3.

Would you like more details or have any questions?

Here are 5 related questions to consider:

1. How do we verify if vectors are collinear in general?
2. What are the geometric interpretations of collinear vectors?
3. How would the problem change if the vectors were parallel but in opposite directions?
4. Can the dot product or cross product be used to check if two vectors are collinear?
5. What are the properties of scalar multiples of vectors?

Tip: Collinear vectors are always parallel, but they could point in the same or opposite directions, depending on the sign of the scalar multiplier.