Which of the following statements is/are true?

(A) The vector sum of the three sides of a triangle in order is $\vec 0$
(B) The magnitude ($r$), direction ratios ($a, b, c$) and direction cosines ($l, m, n$) of any vector $\vec  r= a\hat i + b\hat j + c\hat k$ are related as $l=\frac{a}{r},m=\frac{b}{r},n=\frac{c}{r}$
(C) If θ is the angle between two vectors $\vec a$ and $\vec b$ then their cross product is given as $\vec a ×\vec b = |\vec a||\vec b|\sin θ$
(D) The cross product of two vectors is commutative

Choose the correct answer from the options given below:

(A) and (B) only

The correct answer is Option (3) → (A) and (B) only

(A) True.

For a triangle with consecutive side vectors $\vec{u},\vec{v},\vec{w}$ taken around the triangle, $\vec{u}+\vec{v}+\vec{w}=\vec{0}$.

(B) True.

For $\vec{r}=a\hat{i}+b\hat{j}+c\hat{k}$ with $r=|\vec{r}|=\sqrt{a^{2}+b^{2}+c^{2}}$, direction cosines are $l=\frac{a}{r},\ m=\frac{b}{r},\ n=\frac{c}{r}$.

(C) False.

$|\vec{a}\times\vec{b}|=|\vec{a}|\,|\vec{b}|\sin\theta$, but $\vec{a}\times\vec{b}$ is a vector (magnitude and direction), so the equality $\vec{a}\times\vec{b}=|\vec{a}|\,|\vec{b}|\sin\theta$ (as a vector equality) is incorrect.

(D) False.

Cross product is anti-commutative: $\vec{a}\times\vec{b}=-\,\vec{b}\times\vec{a}$, not commutative.

Correct: (A) and (B)