Question

If $\mathbf{i},\mathbf{j},\mathbf{k}$ are unit orthonormal vectors and a is a vector, if $\mathbf{a} \times \mathbf{r} = \mathbf{j},$ then a . r is

• A. 0
• B. 1
• C. – 1
• D. Arbitrary scalar

Answer 1

Answer: (D)

Arbitrary scalar

Answer 2

Since $|\mathbf{a} \times \mathbf{r}|^{2} + |\mathbf{a}.\mathbf{r}|^{2} = |\mathbf{a}|^{2}|\mathbf{r}|^{2}$

$\Rightarrow |\mathbf{j}|^{2} + (\mathbf{a}.\mathbf{r})^{2} = |\mathbf{a}|^{2}|\mathbf{r}|^{2} \Rightarrow (\mathbf{a}.\mathbf{r}) = \pm \sqrt{|\mathbf{a}|^{2}|\mathbf{r}|^{2} - 1}$

This shows that $\mathbf{a}.\mathbf{r}$ depends on $|\mathbf{r}|$ for given $\mathbf{a}.$

Hence $\mathbf{a}.\mathbf{r}$ is arbitrary scalar.