Which of the following statements are true?
(A) If $\vec r = x\hat i+y\hat j + z\hat k$, 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) $\vec a⊥\vec b$ if and only if $\vec a×\vec b = \vec 0$
(D) Projection of $\vec b$ on $\vec a$ is $\frac{\vec a.\vec b}{|\vec a|^2}$
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) $\vec r = x\hat i + y\hat j + z\hat k$ → $x, y, z$ are **direction ratios**. ✔ True
(B) Vector addition is commutative: $\vec a + \vec b = \vec b + \vec a$. ✔ True
(C) $\vec a \perp \vec b$ ⇔ $\vec a \cdot \vec b = 0$, NOT $\vec a \times \vec b = 0$ (that means vectors are parallel). ✘ False
(D) Projection of $\vec b$ on $\vec a$ is $\frac{\vec a \cdot \vec b}{|\vec a|}$ (a scalar), not $\frac{\vec a \cdot \vec b}{|\vec a|^{2}}$ (that gives a scaling factor, not projection). ✘ False
Correct statements: A and B.