Question

If \( \vec{a}=2\hat{i}-\hat{j}+3\hat{k} \) and \( \vec{b}=\hat{i}+2\hat{j}-\hat{k} \), then the vector perpendicular to both \( \vec a \) and \( \vec b \) is:

• A. \( -5\hat{i}+5\hat{j}+5\hat{k} \)
• B. \( -5\hat{i}+5\hat{j}-5\hat{k} \)
• C. \( 5\hat{i}+5\hat{j}+5\hat{k} \)
• D. \( -5\hat{i}-5\hat{j}+5\hat{k} \)

Hint:

The cross product of two vectors always gives a vector perpendicular to both. Use determinant expansion carefully and remember the middle term carries a negative sign.

Answer 1

The Correct Option is A

To find the vector perpendicular to both \( \vec{a} \) and \( \vec{b} \), we can use the cross product of the two vectors. Given:

• \( \vec{a}=2\hat{i}-\hat{j}+3\hat{k} \)
• \( \vec{b}=\hat{i}+2\hat{j}-\hat{k} \)

The cross product \( \vec{a} \times \vec{b} \) is calculated using the determinant of the following matrix:

\(\hat{i}\)	\(\hat{j}\)	\(\hat{k}\)
2	-1	3
1	2	-1

Now, we compute the determinant by expanding it:

• First component (\(\hat{i}\)): \( (-1)(-1) - (3)(2) = 1 - 6 = -5 \)
• Second component (\(\hat{j}\)): \( -\left[ (2)(-1) - (3)(1) \right] = -( -2 - 3 ) = -(-5) = 5 \)
• Third component (\(\hat{k}\)): \( (2)(2) - (-1)(1) = 4 + 1 = 5 \)

Thus, the cross product, which is the vector perpendicular to both \(\vec{a}\) and \(\vec{b}\), is:

\[ \vec{a} \times \vec{b} = -5\hat{i} + 5\hat{j} + 5\hat{k} \]

Hence, the vector perpendicular to both \( \vec{a} \) and \( \vec{b} \) is \(-5\hat{i}+5\hat{j}+5\hat{k}\), which matches the given correct option.