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

Concept: The vector perpendicular to both vectors is obtained using the cross product: \[ \vec{a}\times \vec{b} \] If: \[ \vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k} \] and \[ \vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k} \] then: \[ \vec{a}\times\vec{b}= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} \]

Step 1: Writing vectors in component form.
Given: \[ \vec{a}=2\hat{i}-\hat{j}+3\hat{k} \] \[ \vec{b}=\hat{i}+2\hat{j}-\hat{k} \] Thus: \[ \vec{a}=(2,-1,3) \] and \[ \vec{b}=(1,2,-1) \]

Step 2: Evaluating cross product.
\[ \vec{a}\times\vec{b}= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -1 & 3 \\ 1 & 2 & -1 \end{vmatrix} \] Expanding determinant: Coefficient of \(\hat{i}\): \[ (-1)(-1)-(3)(2)=1-6=-5 \] Coefficient of \(\hat{j}\): \[ -\left[(2)(-1)-(3)(1)\right] \] \[ =-(-2-3)=5 \] Coefficient of \(\hat{k}\): \[ (2)(2)-(-1)(1)=4+1=5 \] Thus: \[ \vec{a}\times\vec{b}=-5\hat{i}+5\hat{j}+5\hat{k} \]

Step 3: Verifying perpendicularity.
Check dot product with \(\vec{a}\): \[ (-5)(2)+(5)(-1)+(5)(3) =-10-5+15=0 \] Check with \(\vec{b}\): \[ (-5)(1)+(5)(2)+(5)(-1) =-5+10-5=0 \] Hence the vector is perpendicular to both vectors. Therefore: \[ \boxed{-5\hat{i}+5\hat{j}+5\hat{k}} \]