If \( \vec{a} \) and \( \vec{b} \) are two non-zero vectors such that \( \left|\vec{a}\times\vec{b}\right|^{2}+\left|\vec{a}\,.\,\vec{b}\right|^{2}=100 \) and \( |\vec{a}|=5 \) then the value of \( \left|\vec{b}\right| \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 2
With \( \theta \) the angle between the vectors, this is Lagrange's identity:
\( \displaystyle |\vec{a}\times\vec{b}|^2+(\vec{a}\cdot\vec{b})^2=|\vec{a}|^2|\vec{b}|^2\sin^2\theta+|\vec{a}|^2|\vec{b}|^2\cos^2\theta \)
\( \displaystyle =|\vec{a}|^2|\vec{b}|^2 \)
So
\( \displaystyle 25\,|\vec{b}|^2=100 \)
\( \displaystyle |\vec{b}|^2=4 \)
\( \displaystyle |\vec{b}|=2 \)