Question

A vector \( \vec{A} \) when added to the sum of the vectors \( (\hat{i}-2\hat{j}+2\hat{k}) \) and \( (-2\hat{i}+\hat{j}-\hat{k}) \) gives a unit vector along y-axis. The magnitude of vector \( \vec{A} \) is

• A. \( \sqrt{3} \)
• B. \( \sqrt{6} \)
• C. \( \sqrt{8} \)
• D. \( \sqrt{10} \)

Hint:

Always isolate the unknown vector before calculating its magnitude.

Answer 1

The Correct Option is B

Step 1: Add the given vectors.
\[ (\hat{i}-2\hat{j}+2\hat{k}) + (-2\hat{i}+\hat{j}-\hat{k}) = -\hat{i}-\hat{j}+\hat{k} \]
Step 2: Resultant including vector \( \vec{A} \).
\[ \vec{A} + (-\hat{i}-\hat{j}+\hat{k}) = \hat{j} \]
Step 3: Find vector \( \vec{A} \).
\[ \vec{A} = \hat{j} + \hat{i} + \hat{j} - \hat{k} = \hat{i}+2\hat{j}-\hat{k} \]
Step 4: Find magnitude of \( \vec{A} \).
\[ |\vec{A}| = \sqrt{1^2 + 2^2 + (-1)^2} = \sqrt{6} \]
Step 5: Conclusion.
The magnitude of vector \( \vec{A} \) is \( \sqrt{6} \).