Question

If $\overrightarrow{a}=\hat{i}-2\hat{j}+3\hat{k},\overrightarrow{b}=2\hat{i}+3\hat{j}-\hat{k}$ and $\overrightarrow{c}=r\hat{i}+\hat{j}+(2r-1)\hat{k}$ are three vectors such that $\overrightarrow{c}$ is parallel to the plane of $\overrightarrow{a}$ and $\overrightarrow{b}$, then r is equal to?

• A. $0$
• B. $2$
• C. $-1$
• D. $1$

Answer 1

The correct option is A $0$

Given , $\begin{array}{c}\overline{a}=\hat{i}-2\hat{j}+3\hat{k}\\ \overline{b}=2\hat{i}+3\hat{j}+\hat{k}\\ \overline{c}=r\hat{i}+\hat{j}+(2r-1)\hat{k}\end{array}$

also given that

$$\begin{array}{c}\overline{c}isparallelto\overline{a}and\overline{b}\\ letusfindthevectortoboth\overline{a}and\overline{b}\end{array}$$

$\overline{a}\times \overline{b}=\left|\begin{array}{c}\hat{i}\hat{j}\hat{k}\\ 1-23\\ 23-1\end{array}\right|$

$\begin{array}{c}=\hat{i}\left(2-9\right)-\hat{j}\left(-1-6\right)+\hat{k}\left(3+4\right)\\ \overline{a}\times \overline{b}=-7\hat{i}+7\hat{j}+7\hat{k}\end{array}$

$rlinecisparallelto\overline{a}and\overline{b}$

$\overline{c}willparpendicularto\overline{a}\times \overline{b}$

$\overline{c}\cdot \left(\overline{a}\times \overline{b}\right)=\overline{0}$

$\left(r\hat{i}+\hat{j}+(2r-1)\hat{k}\right)\cdot \left(-7\hat{i}+7\hat{j}+7\hat{k}\right)=\overline{0}$

r = 0