The area of triangle with vertices P, Q, R is given by (where \(\overrightarrow{AB}\) = position vector of point B − position vector of point A)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{1}{2}|\overrightarrow{PQ} \times \overrightarrow{PR}|\)
The sides PQ and PR of the triangle meet at P, at angle P. By the definition of the cross product,
\(|\overrightarrow{PQ} \times \overrightarrow{PR}| = PQ \cdot PR\,\sin P\)
The area of a triangle is half the product of two sides and the sine of the included angle:
\(\displaystyle \text{Area} = \frac{1}{2}\,PQ \cdot PR\,\sin P\)
\(\displaystyle = \frac{1}{2}|\overrightarrow{PQ} \times \overrightarrow{PR}|\)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{1}{8}|\overrightarrow{PR} \times \overrightarrow{PQ}|\) — Incorrect. \(|\overrightarrow{PR} \times \overrightarrow{PQ}| = |\overrightarrow{PQ} \times \overrightarrow{PR}|\), so this is only one quarter of the area.
Option (2) \(\displaystyle\frac{1}{4}|\overrightarrow{PR} \times \overrightarrow{PQ}|\) — Incorrect. This is half of the area.
Option (4) \(|\overrightarrow{PQ} \times \overrightarrow{PR}|\) — Incorrect. This is the area of the parallelogram with sides PQ and PR, twice the area of the triangle.
NCERT (2026-27 edition):
Vector (or Cross) Product of Two Vectors
