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) The vector sum of the three sides of a triangle in order is $\vec 0$ . True. 

If we represent the sides of a triangle as vectors $\vec{AB}$, $\vec{BC}$, and $\vec{CA}$ taken in a cyclic order, the sum is $\vec{AB} + \vec{BC} + \vec{CA}$. By the triangle law of vector addition, $\vec{AB} + \vec{BC} = \vec{AC}$. Therefore, $\vec{AC} + \vec{CA} = \vec{AC} - \vec{AC} = \vec{0}$.

(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 $l=\frac{a}{r},m=\frac{b}{r},n=\frac{c}{r}$.True.

For a vector $\vec{r} = a\hat{i} + b\hat{j} + c\hat{k}$, the direction ratios are $(a, b, c)$ and the magnitude is $r = \sqrt{a^2 + b^2 + c^2}$. The direction cosines are defined as the ratio of the components to the magnitude: $l = \frac{a}{r}, m = \frac{b}{r}, n = \frac{c}{r}$

(C) If θ is the angle between two vectors $\vec a$ and $\vec b$ then their cross product is given as $\vec a ×\vec b = |\vec a||\vec b|\sin θ$ . False.

The cross product $\vec{a} \times \vec{b}$ is a vector, not a scalar. The correct formula is:

where $\hat{n}$ is the unit vector perpendicular to both $\vec{a}$ and $\vec{b}$. The expression $|\vec{a}||\vec{b}| \sin \theta$ only gives the magnitude of the cross product.

(D) The cross product of two vectors is commutative. False.

The cross product is anti-commutative. This means:

Changing the order of the vectors reverses the direction of the resulting vector (Right-hand thumb rule).