Question

If $\vec{\left|c\right|^{2}} = 60$ and $\vec{c}\times\left(\hat{i}+2\hat{j} + 5 \hat{k}\right) = \vec{0},$ then a value of $\vec{c} \cdot \left(-7 \hat{i} + 2\hat{j} +3\hat{k}\right)$ is :

• A. $4\sqrt{2}$
• B. 12
• C. 24
• D. $12\sqrt{2}$

Answer 1

Answer: (D)

$\bar{C}\times\left(\hat{i}+2\hat{j} + 5 \hat{k}\right) = 0$
$\bar{C}=\lambda \left(\hat{i}+2\hat{j} + 5 \hat{k}\right)$
$\left|C\right| = \lambda \sqrt{30} \Rightarrow \lambda^{2} \left(30\right) = \left|c\right|^{2} = 60$
$\lambda = \pm \sqrt{2}$
$\Rightarrow \bar{C}. \left(-7\hat{i}+2\hat{j} + 3 \hat{k}\right)$
$\Rightarrow \lambda \left(\hat{i}+2\hat{j} + 5 \hat{k}\right).\left(-7\hat{i}+2\hat{j} + 3 \hat{k}\right)$
$\Rightarrow \lambda \left(-7+4 + 15\right) = 12\lambda$
$= 12\sqrt{2}\quad$ or $-12\sqrt{2}$