If \( \vec{a}=2\hat{\imath}-2\hat{\jmath}+\hat{k} \), then which of the following statements are TRUE?
(A) unit vector along \( \vec{a} \) is \( \displaystyle\frac{1}{3}(2\hat{\imath}+2\hat{\jmath}-\hat{k}) \)
(B) \( |3\vec{a}|=9 \)
(C) \( \vec{a} \) is perpendicular to \( 2\hat{\imath}+3\hat{\jmath}+2\hat{k} \)
(D) Direction Ratios of \( \vec{a} \) are 2,−2,1
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B), (C) and (D) only
\( |\vec{a}|=\sqrt{2^{2}+(-2)^{2}+1^{2}} \)
\( =3 \)
The unit vector along \( \vec{a} \) is
\( \displaystyle \hat{a}=\frac{1}{3}(2\hat{\imath}-2\hat{\jmath}+\hat{k}) \)
The dot product with \( 2\hat{\imath}+3\hat{\jmath}+2\hat{k} \) is
\( \vec{a}\cdot(2\hat{\imath}+3\hat{\jmath}+2\hat{k})=4-6+2 \)
\( =0 \)
So \( |\vec{a}|=3 \), the unit vector along \( \vec{a} \) is \( \frac{1}{3}(2\hat{\imath}-2\hat{\jmath}+\hat{k}) \), \( \vec{a} \) is perpendicular to \( 2\hat{\imath}+3\hat{\jmath}+2\hat{k} \), and the direction ratios of \( \vec{a} \) are its components 2, −2, 1.
Checking the statements
(A) unit vector along \( \vec{a} \) is \( \frac{1}{3}(2\hat{\imath}+2\hat{\jmath}-\hat{k}) \) — Incorrect. The unit vector is \( \frac{1}{3}(2\hat{\imath}-2\hat{\jmath}+\hat{k}) \); the signs of the \( \hat{\jmath} \) and \( \hat{k} \) terms are reversed.
(B) \( |3\vec{a}|=9 \) — Correct. \( |3\vec{a}|=3\times3=9 \).
(C) \( \vec{a} \) is perpendicular to \( 2\hat{\imath}+3\hat{\jmath}+2\hat{k} \) — Correct. The dot product is 4 − 6 + 2 = 0.
(D) Direction Ratios of \( \vec{a} \) are 2, −2, 1 — Correct. These are the components of \( \vec{a} \).