If \(\left(\vec{a}-\vec{b}\right)\cdot\left(\vec{a}+\vec{b}\right)=27\) and \(|\vec{a}|=2|\vec{b}|\), then \(|\vec{b}|\) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 3
Expanding the dot product, the middle terms cancel because \(\vec{a}\cdot\vec{b} = \vec{b}\cdot\vec{a}\):
\(\left(\vec{a}-\vec{b}\right)\cdot\left(\vec{a}+\vec{b}\right) = |\vec{a}|^2 - |\vec{b}|^2\)
So, with \(|\vec{a}| = 2|\vec{b}|\):
\(4|\vec{b}|^2 - |\vec{b}|^2 = 27\)
\(3|\vec{b}|^2 = 27\)
\(|\vec{b}|^2 = 9\)
\(|\vec{b}| = 3\)