Question

\(\bar{a} = \hat{i} - \hat{j}, \bar{b} = \hat{j} - \hat{k}, \bar{c} = \hat{k} - \hat{i}\) then a unit vector \(\bar{d}\) such that \(\bar{a} \cdot \bar{d} = 0 = [\bar{b} \bar{c} \bar{d}]\) is

• A. \(\pm \left( \frac{\hat{i} + \hat{j} + 3\hat{k}}{\sqrt{11}} \right)\)
• B. \(\pm \left( \frac{-\hat{j} + \hat{k}}{\sqrt{2}} \right)\)
• C. \(\pm \left( \frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}} \right)\)
• D. \(\pm \left( \frac{\hat{i} + \hat{j} - 2\hat{k}}{\sqrt{6}} \right)\)

Hint:

The vector d must be perpendicular to a and must make the scalar triple product with b and c zero.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Let \(\bar{d} = x\hat{i} + y\hat{j} + z\hat{k}\). Then \(\bar{a}\cdot\bar{d} = x - y = 0\), so \(x = y\).

Step 2: Triple Product
\([\bar{b}\ \bar{c}\ \bar{d}] = \bar{d}\cdot(\bar{b}\times\bar{c})\). Here \(\bar{b}\times\bar{c} = \hat{i} + \hat{j} + \hat{k}\), so we need \(x + y + z = 0\).

Step 3: Solve
With \(x = y\): \(z = -2x\). So \(\bar{d} \propto \hat{i} + \hat{j} - 2\hat{k}\), and its magnitude is \(\sqrt{6}\).

Step 4: Compare the options
The unit vector is \(\pm\dfrac{\hat{i} + \hat{j} - 2\hat{k}}{\sqrt{6}}\), the last option.

Final Answer:
The unit vector is plus or minus (i + j - 2k) divided by root 6. \[ \boxed{\text{(D) }\pm\left(\dfrac{\hat{i}+\hat{j}-2\hat{k}}{\sqrt{6}}\right)} \]