Which of the following statements is/are true? (A) The vector sum of…

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

(A) The vector sum of the three sides of a triangle in order is \(\vec{0}\) — Correct. For a triangle ABC, the triangle law gives \(\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}\), so

    \(\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA} = \overrightarrow{AC} + \overrightarrow{CA}\)
    \(\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA} = \overrightarrow{AC} - \overrightarrow{AC} = \vec{0}\)

(B) The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of any vector \(\vec{r} = a\hat{\imath} + b\hat{\jmath} + c\hat{k}\) are related as \(l=\frac{a}{r}, m=\frac{b}{r}, n=\frac{c}{r}\) — Correct. Here \(r = \sqrt{a^2 + b^2 + c^2}\), and the direction cosines are the components of the unit vector \(\frac{\vec{r}}{r}\).

(C) If \(\theta\) is the angle between two vectors \(\vec{a}\) and \(\vec{b}\) then their cross product is given as \(\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta\) — Incorrect. The cross product is a vector:

    \(\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta\;\hat{n}\)

where \(\hat{n}\) is the unit vector perpendicular to both \(\vec{a}\) and \(\vec{b}\). The number \(|\vec{a}||\vec{b}|\sin\theta\) is only its magnitude.

(D) The cross product of two vectors is commutative — Incorrect. Reversing the order reverses the direction (right-hand rule), so the cross product is anti-commutative:

    \(\vec{b} \times \vec{a} = -(\vec{a} \times \vec{b})\)

So only (A) and (B) are true.