Which of the following statements are true?
(A) If \(\vec r = x\hat{\imath}+y\hat{\jmath} + z\hat{k}\), then \(x, y, z\) are called direction ratios of \(\vec r\).
(B) For any two vectors \(\vec a\) and \(\vec b\), \(\vec a+\vec b= \vec b+\vec a\)
(C) \(\vec a\perp\vec b\) if and only if \(\vec a\times\vec b = \vec 0\)
(D) Projection of \(\vec b\) on \(\vec a\) is \(\displaystyle\frac{\vec a.\vec b}{|\vec a|^2}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (B) only
(A) If \(\vec r = x\hat{\imath}+y\hat{\jmath} + z\hat{k}\), then \(x, y, z\) are called direction ratios of \(\vec r\). — Correct. The direction cosines of \(\vec r\) are \(\frac{x}{r}, \frac{y}{r}, \frac{z}{r}\) with \(r = |\vec r|\), and x, y, z are proportional to them, so they are direction ratios.
(B) For any two vectors \(\vec a\) and \(\vec b\), \(\vec a+\vec b= \vec b+\vec a\) — Correct. Vector addition is commutative.
(C) \(\vec a\perp\vec b\) if and only if \(\vec a\times\vec b = \vec 0\) — Incorrect. For non-zero vectors, \(\vec a\times\vec b = \vec 0\) means \(\vec a\) and \(\vec b\) are parallel; \(\vec a\perp\vec b\) if and only if \(\vec a\cdot\vec b = 0\).
(D) Projection of \(\vec b\) on \(\vec a\) is \(\frac{\vec a.\vec b}{|\vec a|^2}\) — Incorrect. The projection of \(\vec b\) on \(\vec a\) is \(\frac{\vec a\cdot\vec b}{|\vec a|}\); the printed expression equals it only when \(|\vec a| = 1\).