Question:
Let \(\bar{a}=2 \hat{i}+\hat{j}+\hat{k}, \bar{b}=\hat{i}+2 \hat{j}-\hat{k}\) and vector \(\bar{c}\) be coplanar. If \(\bar{c}\) is perpendicular to \(\bar{a}\), then \(\bar{c}\) is
Let \(\bar{a}=2 \hat{i}+\hat{j}+\hat{k}, \bar{b}=\hat{i}+2 \hat{j}-\hat{k}\) and vector \(\bar{c}\) be coplanar. If \(\bar{c}\) is perpendicular to \(\bar{a}\), then \(\bar{c}\) is
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The required vector is parallel to a x (a x b), because it lies in the plane of a and b and is perpendicular to a.
Updated On: Nov 13, 2026
- \(-\hat{\mathrm{i}}+2\hat{\mathrm{k}}\)
- \(-\hat{i}+\hat{j}+\hat{k}\)
- \(\hat{\mathrm{i}}-2\hat{\mathrm{j}}\)
- \(-\hat{\mathrm{j}}+\hat{\mathrm{k}}\)
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept
A vector in the plane of \(\bar{a}\) and \(\bar{b}\) that is perpendicular to \(\bar{a}\) is parallel to \(\bar{a}\times(\bar{a}\times\bar{b})\).
Step 2: Compute a x b
\(\bar{a}\times\bar{b} = (-3, 3, 3)\).
Step 3: Compute a x (a x b)
\((2, 1, 1)\times(-3, 3, 3) = (0, -9, 9)\), parallel to \((0, -1, 1)\).
Step 4: Match
The vector \(-\hat{j} + \hat{k}\) fits option D.
Final Answer:
The vector is \(-\hat{j} + \hat{k}\). \[ \boxed{\text{(D) }-\hat{\mathrm{j}}+\hat{\mathrm{k}}} \]
A vector in the plane of \(\bar{a}\) and \(\bar{b}\) that is perpendicular to \(\bar{a}\) is parallel to \(\bar{a}\times(\bar{a}\times\bar{b})\).
Step 2: Compute a x b
\(\bar{a}\times\bar{b} = (-3, 3, 3)\).
Step 3: Compute a x (a x b)
\((2, 1, 1)\times(-3, 3, 3) = (0, -9, 9)\), parallel to \((0, -1, 1)\).
Step 4: Match
The vector \(-\hat{j} + \hat{k}\) fits option D.
Final Answer:
The vector is \(-\hat{j} + \hat{k}\). \[ \boxed{\text{(D) }-\hat{\mathrm{j}}+\hat{\mathrm{k}}} \]
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