Question

If \[ a = \gamma \hat{i} + \delta \hat{j} + 2\zeta \hat{k}, \, b = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}, \, r \times a = b \times a \, r \times b = a \times b \] then a unit vector in the direction of \( r \) is

• A. \( \frac{1}{\gamma \zeta} \left( \hat{i} + 3\hat{j} - \hat{k} \right) \)
• B. \( \frac{1}{\gamma \zeta} \left( \hat{i} + \hat{j} + \hat{k} \right) \)
• C. \( \frac{1}{\sqrt{3}} \left( \hat{i} + \hat{j} + \hat{k} \right) \)
• D. None of these

Hint:

When dealing with cross products in vector algebra, remember the properties of the vector triple product and use the formula to simplify the expression.

Answer 1

The Correct Option is C

Using the vector triple product identity, the unit vector in the direction of \( r \) is given by the normalized form of the resulting vector.
After calculation, the correct unit vector is \( \frac{1}{\sqrt{3}} \left( \hat{i} + \hat{j} + \hat{k} \right) \).