Question:
A vector which is orthogonal to the vector \(\overset{̄}{a} = \hat{i}+2\hat{j}+3\hat{k}\) and coplanar with the vectors \(\overset{̄}{b} = 3\hat{i}+2\hat{j}\) and \(\overset{̄}{c} = 2\hat{i}+\hat{j}+3\hat{k}\) is
A vector which is orthogonal to the vector \(\overset{̄}{a} = \hat{i}+2\hat{j}+3\hat{k}\) and coplanar with the vectors \(\overset{̄}{b} = 3\hat{i}+2\hat{j}\) and \(\overset{̄}{c} = 2\hat{i}+\hat{j}+3\hat{k}\) is
Show Hint
Use the vector triple product, since a x (b x c) lies in the plane of b and c and is perpendicular to a.
Updated On: Oct 1, 2026
- \(25\hat{i}+19\hat{j}-21\hat{k}\)
- \(-25\hat{i}+19\hat{j}-21\hat{k}\)
- \(-25\hat{i}+19\hat{j}+21\hat{k}\)
- \(25\hat{i}+19\hat{j}+21\hat{k}\)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept:
The vector \(\bar a\times(\bar b\times\bar c)\) is perpendicular to \(\bar a\) and also lies in the plane of \(\bar b\) and \(\bar c\). This is exactly what the question asks for.
Step 2: Key Formula or Approach:
\(\bar a\times(\bar b\times\bar c) = (\bar a\cdot\bar c)\bar b - (\bar a\cdot\bar b)\bar c\).
Step 3: Detailed Explanation:
\(\bar a\cdot\bar c = 2 + 2 + 9 = 13\) and \(\bar a\cdot\bar b = 3 + 4 + 0 = 7\).
\(13\bar b - 7\bar c = 13(3, 2, 0) - 7(2, 1, 3) = (39 - 14,\ 26 - 7,\ -21) = (25, 19, -21)\).
\[ 25\hat i + 19\hat j - 21\hat k \]
Check orthogonality with \(\bar a\): \(25 + 38 - 63 = 0\).
Final Answer:
The vector is \(25\hat i + 19\hat j - 21\hat k\), option (A). \[ \boxed{25\hat{i}+19\hat{j}-21\hat{k}} \]
The vector \(\bar a\times(\bar b\times\bar c)\) is perpendicular to \(\bar a\) and also lies in the plane of \(\bar b\) and \(\bar c\). This is exactly what the question asks for.
Step 2: Key Formula or Approach:
\(\bar a\times(\bar b\times\bar c) = (\bar a\cdot\bar c)\bar b - (\bar a\cdot\bar b)\bar c\).
Step 3: Detailed Explanation:
\(\bar a\cdot\bar c = 2 + 2 + 9 = 13\) and \(\bar a\cdot\bar b = 3 + 4 + 0 = 7\).
\(13\bar b - 7\bar c = 13(3, 2, 0) - 7(2, 1, 3) = (39 - 14,\ 26 - 7,\ -21) = (25, 19, -21)\).
\[ 25\hat i + 19\hat j - 21\hat k \]
Check orthogonality with \(\bar a\): \(25 + 38 - 63 = 0\).
Final Answer:
The vector is \(25\hat i + 19\hat j - 21\hat k\), option (A). \[ \boxed{25\hat{i}+19\hat{j}-21\hat{k}} \]
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