If \( \vec{a}=2\hat{\imath}+\hat{\jmath} \) , and \( \vec{b}=\hat{\jmath}-\hat{k} \) , then which of the following statements is/are TRUE?
A. \( |\vec{a}+\vec{b}|=3 \)
B. Unit vector along \( \vec{a}+\vec{b} \) is \( 2\hat{\imath}+2\hat{\jmath}-\hat{k} \)
C. Direction ratio of \( \vec{a}+\vec{b} \) are 2, 2, −1
D. unit vector perpendicular to \( \vec{a} \) and \( \vec{b} \) is \( -\displaystyle\frac{1}{3}\hat{\imath}+\displaystyle\frac{2}{3}\hat{\jmath}+\displaystyle\frac{2}{3}\hat{k} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A, C and D only
\( \displaystyle \vec{a}+\vec{b}=2\hat{\imath}+2\hat{\jmath}-\hat{k} \)
\( \displaystyle |\vec{a}+\vec{b}|=\sqrt{4+4+1}=3 \)
A vector perpendicular to both is the cross product:
\( \displaystyle \vec{a}\times\vec{b}=\begin{vmatrix}\hat{\imath} & \hat{\jmath} & \hat{k}\\2 & 1 & 0\\0 & 1 & -1\end{vmatrix} \)
\( \displaystyle =(-1-0)\hat{\imath}-(-2-0)\hat{\jmath}+(2-0)\hat{k} \)
\( \displaystyle =-\hat{\imath}+2\hat{\jmath}+2\hat{k} \)
\( \displaystyle |\vec{a}\times\vec{b}|=\sqrt{1+4+4}=3 \)
\( \displaystyle \frac{\vec{a}\times\vec{b}}{3}=-\frac{1}{3}\hat{\imath}+\frac{2}{3}\hat{\jmath}+\frac{2}{3}\hat{k} \)
So \( \vec{a}+\vec{b}=2\hat{\imath}+2\hat{\jmath}-\hat{k} \) with magnitude 3, and a unit vector perpendicular to both is \( -\frac{1}{3}\hat{\imath}+\frac{2}{3}\hat{\jmath}+\frac{2}{3}\hat{k} \).
Checking the statements
(A) \( |\vec{a}+\vec{b}|=3 \) — Correct. \( \sqrt{4+4+1}=3 \).
(B) Unit vector along \( \vec{a}+\vec{b} \) is \( 2\hat{\imath}+2\hat{\jmath}-\hat{k} \) — Incorrect. This vector has magnitude 3, not 1; the unit vector is \( \frac{1}{3}(2\hat{\imath}+2\hat{\jmath}-\hat{k}) \).
(C) Direction ratio of \( \vec{a}+\vec{b} \) are 2, 2, −1 — Correct. The components 2, 2, −1 of a vector are a set of its direction ratios.
(D) unit vector perpendicular to \( \vec{a} \) and \( \vec{b} \) is \( -\frac{1}{3}\hat{\imath}+\frac{2}{3}\hat{\jmath}+\frac{2}{3}\hat{k} \) — Correct. It is \( \frac{\vec{a}\times\vec{b}}{3} \), with \( \vec{a}\times\vec{b}=-\hat{\imath}+2\hat{\jmath}+2\hat{k} \).