Which of the following statements are correct?
(A) If $\vec a$ and $\vec b$ represent the adjacent sides of a triangle, then its area is $\frac{1}{2}|\vec a×\vec b|$
(B) If $\vec a$ and $\vec b$ represent the adjacent sides of a parallelogram, then its area is $|\vec a×\vec b|$
(C) $|\vec a×\vec b|=|\vec a||\vec b|\cos θ$
(D) If $\vec a$ and $\vec b$ represent the 'diagonals' of a parallelogram, then its area is is $\frac{1}{2}|\vec a×\vec b|$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B) and (D) only
(A) If $\vec{a}$ and $\vec{b}$ are adjacent sides of a triangle, then area = $\frac{1}{2}|\vec{a} \times \vec{b}|$
✔️ Correct. This is the standard formula for the area of a triangle formed by two vectors.
(B) If $\vec{a}$ and $\vec{b}$ are adjacent sides of a parallelogram, then area = $|\vec{a} \times \vec{b}|$
✔️ Correct. The cross product magnitude gives the area of the parallelogram.
(C) $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\cos\theta$
❌ Incorrect. The correct identity is $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$
(D) If $\vec{a}$ and $\vec{b}$ are diagonals of a parallelogram, then area = $\frac{1}{2}|\vec{a} \times \vec{b}|$
✔️ Correct. This is a standard formula for area of parallelogram when the diagonal vectors are given.
Correct options: (A), (B), (D)