Question

If $\overset{ \rightarrow }{r}\cdot \hat{i}=2\overset{ \rightarrow }{r}\cdot \hat{j}=4\overset{ \rightarrow }{r}\cdot \hat{k}$ and $\left|\right.\overset{ \rightarrow }{r}\left|\right.=\sqrt{84}$ , then the value of $\overset{ \rightarrow }{r}\cdot \left(2 \hat{i} - 3 \hat{j} + \hat{k}\right)$ may be

• A. $0$
• B. $2$
• C. $4$
• D. $6$

Answer 1

Answer: (D)

$\overset{ \rightarrow }{r}\cdot \hat{i}=2\overset{ \rightarrow }{r}\cdot \hat{j}=4\overset{ \rightarrow }{r}\cdot \hat{k}$
Let $\overset{ \rightarrow }{r}=x\hat{i}+y\hat{j}+z\overset{ \rightarrow }{k}$
$x=2y=4z$
$\left|\right.\overset{ \rightarrow }{r}\left|\right.=\sqrt{84}\Rightarrow \sqrt{\left(4 z\right)^{2} + \left(2 z\right)^{2} + z^{2}}=\sqrt{84}$
$z=2;y=4;x=8$
$\overset{ \rightarrow }{r}=8\hat{i}+4\hat{j}+2\hat{k}$
$\overset{ \rightarrow }{r}\cdot \left(2 \hat{i} - 3 \hat{j} + \hat{k}\right)=16 \, -12+2=6$