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. \(25\hat{i}+19\hat{j}-21\hat{k}\)
• B. \(-25\hat{i}+19\hat{j}-21\hat{k}\)
• C. \(-25\hat{i}+19\hat{j}+21\hat{k}\)
• D. \(25\hat{i}+19\hat{j}+21\hat{k}\)

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.

Answer 1

The Correct Option is A

Step 1: Test the options for orthogonality:
Dot each option with $\bar a = (1, 2, 3)$. A: $25 + 38 - 63 = 0$. B: $-25 + 38 - 63 = -50$. C: $-25 + 38 + 63 = 76$. D: $25 + 38 + 63 = 126$. Only A is orthogonal.

Step 2: Confirm coplanarity:
Check that $[\bar d\ \bar b\ \bar c] = 0$ for $\bar d = (25, 19, -21)$: $\bar d = 13\bar b - 7\bar c$ is a combination of $\bar b$ and $\bar c$, so it is coplanar with them.

Final Answer:
The vector is $25\hat i + 19\hat j - 21\hat k$, option (A). \[ \boxed{25\hat{i}+19\hat{j}-21\hat{k}} \]