Let $\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}$ Assertion…

Let $\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}$ Assertion (A) : The identity $|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2$ holds for $\vec{a}$. Reason (R) : $\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}$, $\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}$ Which of the following is correct?
  1. Both $(\mathrm{A})$ and $(\mathrm{R})$ are true and $(\mathrm{R})$ is the correct reason for (A)
  2. Both $(A)$ and $(R)$ are true but $(R)$ is not the correct reason for (A)
  3. (A) is true, (R) is false
  4. (A) is false, (R) is true

Solution

Given, $\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}$ Now, $|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2=(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}) \cdot(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}})$ $\begin{aligned} & =\left(a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\right) \cdot\left(a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\right) \\ & =a_3^2+a_2^2\end{aligned}$ $\begin{aligned}|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2 & =\left(a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}\right) \cdot\left(a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}\right) \\ & =a_1^2+a_3^2\end{aligned}$ $\begin{aligned}|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2 & =\left(a_1 \hat{\mathbf{i}}+a_1 \hat{\mathbf{j}}\right) \cdot\left(a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\right) \\ & =a_2{ }^2+a_1{ }^2\end{aligned}$ Now, $\begin{aligned}|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2 & +|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2 \\ & =2\left(a_1^2+a_2^2+a_3^2\right) \\ & =2|\overrightarrow{\mathbf{a}}|^2\end{aligned}$ Hence, both $\mathrm{A}$ and $\mathrm{R}$ are true and $\mathrm{R}$ is correct reason for $\mathrm{A}$.

Asked in: AP EAMCET 2007

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