Which of the following statements are correct ?
A. If \( \vec{a} \) and \( \vec{b} \) are two adjacent sides of a triangle, then the area of the triangle is \( \displaystyle\frac{1}{2}|\vec{a}\times\vec{b}| \)
B. If \( \vec{a} \) and \( \vec{b} \) are two adjacent sides of a triangle, then the area of the triangle is \( \displaystyle\frac{1}{2}|\vec{a}.\vec{b}| \)
C. If \( \vec{a} \) and \( \vec{b} \) are two adjacent sides of a parallelogram, then its area is \( |\vec{a}\times\vec{b}| \)
D. If \( \vec{a} \) and \( \vec{b} \) are two adjacent sides of a parallelogram, then its area is \( |\vec{a}.\vec{b}| \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → A and C only
The magnitude of the cross product, \( |\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|\sin\theta \), is base × height of the parallelogram with sides \( \vec{a} \) and \( \vec{b} \). The triangle on the same two sides is half of it. The dot product \( |\vec{a}||\vec{b}|\cos\theta \) is not an area: it is zero for perpendicular sides, where the area is largest.
So the triangle has area \( \frac{1}{2}|\vec{a}\times\vec{b}| \) and the parallelogram has area \( |\vec{a}\times\vec{b}| \).
Checking the statements
(A) If \( \vec{a} \) and \( \vec{b} \) are two adjacent sides of a triangle, then the area of the triangle is \( \frac{1}{2}|\vec{a}\times\vec{b}| \) — Correct. The triangle is half the parallelogram.
(B) If \( \vec{a} \) and \( \vec{b} \) are two adjacent sides of a triangle, then the area of the triangle is \( \frac{1}{2}|\vec{a}.\vec{b}| \) — Incorrect. The area uses the cross product, not the dot product.
(C) If \( \vec{a} \) and \( \vec{b} \) are two adjacent sides of a parallelogram, then its area is \( |\vec{a}\times\vec{b}| \) — Correct. \( |\vec{a}||\vec{b}|\sin\theta \) is base × height.
(D) If \( \vec{a} \) and \( \vec{b} \) are two adjacent sides of a parallelogram, then its area is \( |\vec{a}.\vec{b}| \) — Incorrect. The area is \( |\vec{a}\times\vec{b}| \); the dot product is zero for perpendicular sides.