Question

If $\overrightarrow{a}=-\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=2\hat{i}+0\hat{j}+\hat{k}$ then the vector $\overrightarrow{c}$ satisfying the condition, (i) that it is coplanar with $\overrightarrow{a}\&\overrightarrow{b}$ (ii) that it is perpendicular to $\overrightarrow{b}$, and (iii) that $\overrightarrow{a}\cdot \overrightarrow{c}=7$, is-

• A. $-\frac{3}{2}\hat{i}+\frac{5}{2}\hat{j}+3\hat{k}$
• B. $-3i+5\hat{j}+6\hat{k}$
• C. $-6\hat{i}+0\hat{j}+5\hat{k}$
• D. $-\hat{i}+2\hat{j}+2\hat{k}$

Answer 1

The correct option is B $-3i+5\hat{j}+6\hat{k}$

If $\overrightarrow{a}=-\hat{i}+\hat{j}+\hat{k}\overrightarrow{b}=2\hat{i}+0\hat{j}+\hat{k}$

$\left(i\right)$ Checking the coplanarity

$\overrightarrow{a}.\overrightarrow{b}0\cos \theta =0\because \theta =90°\overrightarrow{b}.\overrightarrow{c}=0,\overrightarrow{c}.\overrightarrow{a}=0(2\hat{i}+0\hat{j}+\hat{j})(-3\hat{i}+5\hat{j}+6\hat{k})=-6+0+6=0$

$\left(ii\right)$ Checking it is perpendicular to

$\overrightarrow{b}\overrightarrow{b}\times \overrightarrow{c}=1\sin 90°=1\overrightarrow{b}.\overrightarrow{c}=0\cos 90°=0$

It follows option $\left(B\right)$ as well