Question

For any vector a, the value of ${(a\times i)}^2+{(a\times k)}^2$ is equal to

• A. $3a^{\wedge }2$
• B. $a^{\wedge }2$
• C. $\left(a_1^2+2a_2^2+a_3^2\right)$
• D. $4a^{\wedge }2$
• E. Answer Required

Answer 1

Answer: (C)

$\left(a_1^2+2a_2^2+a_3^2\right)$

$\begin{array}{c}{\left(a\times i\right)}^2+{\left(a\times k\right)}^2\\ Let\\ \overrightarrow{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\\ ={\left\{\left(a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\right)\times \hat{i}\right\}}^2+{\left\{\left(a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\right)\times \hat{k}\right\}}^2\\ ={\left(a_3\hat{i}-a_2\hat{k}\right)}^2+{\left(a_2\hat{i}-a_1\hat{j}\right)}^2\\ =a_3^2+a_2^2+a_2^2+a_1^2\\ =\left(a_1^2+2a_2^2+a_3^2\right)\\ Hence,\\ optionCiscorrectanswer.\end{array}$