Question

If $ \vec{a} = \hat{i} + \hat{j}, \vec{b} = 2\hat{j} - \hat{k} $ are two vectors such that $ \vec{r} \times \vec{a} = \vec{b} \times \vec{a}, \vec{r} \times \vec{b} = \vec{a} \times \vec{b} $, then the unit vector in the direction of $ \vec{r} $ is:

• A. \( \frac{1}{\sqrt{11}}(\hat{i} + 3\hat{j} - \hat{k}) \)
• B. \( \frac{1}{\sqrt{11}}(\hat{i} - 3\hat{j} + \hat{k}) \)
• C. \( \frac{1}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k}) \)
• D. \( \frac{1}{\sqrt{3}}(\hat{i} + \hat{j} - \hat{k}) \)

Hint:

Use vector identities like \( \vec{a} \times \vec{a} = 0 \), and parallel vector conditions to set up linear equations.

Answer 1

The Correct Option is A

We are given: \[ \vec{r} \times \vec{a} = \vec{b} \times \vec{a} \Rightarrow (\vec{r} - \vec{b}) \times \vec{a} = 0 \Rightarrow \vec{r} - \vec{b} \parallel \vec{a} \Rightarrow \vec{r} = \vec{b} + \lambda \vec{a} \] Also, \[ \vec{r} \times \vec{b} = \vec{a} \times \vec{b} \Rightarrow (\vec{r} - \vec{a}) \times \vec{b} = 0 \Rightarrow \vec{r} - \vec{a} \parallel \vec{b} \Rightarrow \vec{r} = \vec{a} + \mu \vec{b} \] Solve both expressions for \( \vec{r} \), equate: \[ \vec{b} + \lambda \vec{a} = \vec{a} + \mu \vec{b} \] Substitute and solve to find \( \vec{r} = \hat{i} + 3\hat{j} - \hat{k} \) Unit vector is: \[ \hat{r} = \frac{1}{\sqrt{1^2 + 3^2 + (-1)^2}}(\hat{i} + 3\hat{j} - \hat{k}) = \frac{1}{\sqrt{11}}(\hat{i} + 3\hat{j} - \hat{k}) \]