Question

If $\vec{a} =\left(1, p, 1\right)$, $\vec{b} =\left(q,2,2\right)$, $\vec{a}\cdot\vec{b} = r$ and $\vec{a}\times \vec{b} =\left(0,-3,3\right)$, then find the value of $p$, $q$, $r$ respectively.

• A. $1$, $5$, $9$
• B. $9$, $5$, $1$
• C. $5$, $1$, $9$
• D. None of these

Answer 1

Answer: (D)

Given, $\vec{a}\cdot\vec{b} = r$
$\Rightarrow \left(\hat{i}+p\hat{j}+\hat{k}\right)\cdot \left(q\hat{i}+2\hat{j}+2\hat{k}\right) = r$
$\Rightarrow q + 2p + 2 = r \quad\ldots\left(i\right)$
and $\vec{a}\times \vec{b} = 0\hat{i}-3\hat{j}+3\hat{k}$
$\Rightarrow \left(2p-2\right)\hat{i}+\hat{j}\left(q-2\right)+\hat{k}\left(2-pq\right) = 0\hat{i}-3\hat{j}+3\hat{k}$
$\Rightarrow 2p - 2 = 0$; $q - 2 = -3$ and $2 -pq = 3$
$\Rightarrow p = 1$, $q = - 1$
$\therefore $ From eq. $\left(i\right)$, $r = 3$