Question

If $a=2\hat{i}+3\hat{j}-\hat{k},b=\hat{i}+2\hat{j}-5\hat{k},c=3\hat{i}+5\hat{j}-\hat{k},$ then a vector perpendicular to $a$ and in the plane containing $b$and $c$ is$?$

• A. $-17\hat{i}+21\hat{j}–97\hat{k}$
• B. $17\hat{i}+21\hat{j}–123\hat{k}$
• C. $-17\hat{i}-21\hat{j}+97\hat{k}$
• D. $-17\hat{i}-21\hat{j}-97\hat{k}$

Answer 1

The correct option is D

$-17\hat{i}-21\hat{j}-97\hat{k}$

Find the vector perpendicular to $a$ and in the plane containing $b$and $c$ :

Given that the vectors $a=2\hat{i}+3\hat{j}-\hat{k},b=\hat{i}+2\hat{j}-5\hat{k},c=3\hat{i}+5\hat{j}-\hat{k},$

Let the required vector be $\overrightarrow{r}$

As per the given condition

$$\begin{gathered} (\overrightarrow{b}\times \overrightarrow{c})=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& -5\\ 3& 5& -1\end{array}\right| \\ \Rightarrow (\overrightarrow{b}\times \overrightarrow{c})=\hat{i}(-2+25)-\hat{j}(-1+15)+\hat{k}(5-6) \\ \Rightarrow (\overrightarrow{b}\times \overrightarrow{c})=23\hat{i}-14\hat{j}-\hat{k} \end{gathered}$$

$$\begin{gathered} \overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 3& -1\\ 23& -14& -1\end{array}\right| \\ \Rightarrow \overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\hat{i}(-3-14)-\hat{j}(-2+23)+\hat{k}(-28-69) \\ \Rightarrow \overrightarrow{a}\times \overrightarrow{(b}\times \overrightarrow{c})=-17\hat{i}-21\hat{j}-97\hat{k} \end{gathered}$$

Hence, the correct option is (D).