If \(\vec{a}\) and \(\vec{b}\) are two vectors such that \(|\vec{a}| = 10, |\vec{b}| = 2\) and \(\vec{a}.\vec{b} = 12\), then \(|\vec{a} \times \vec{b}|\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 16
The dot product gives the angle \(\theta\) between the vectors:
\(\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta\)
\(12 = 10 \times 2 \times \cos\theta\)
\( \displaystyle \cos\theta = \frac{12}{20} = \frac{3}{5} \)
Since \(0 \le \theta \le \pi\), \(\sin\theta \ge 0\):
\( \displaystyle \sin\theta = \sqrt{1 - \frac{9}{25}} = \frac{4}{5} \)
The magnitude of the cross product is
\(|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta\)
\( \displaystyle = 10 \times 2 \times \frac{4}{5} = 16 \)
NCERT Exemplar:
Vector Algebra
