Question

If \((\vec a+3\vec b)\) is perpendicular to \((7\vec a-5\vec b)\) and \((\vec a-4\vec b)\) is perpendicular to \((7\vec a-2\vec b)\), then the angle between \(\vec a\) and \(\vec b\) (in degrees) is ____________.

Hint:

Set both dot products to zero and solve for \(|\vec a|^2\) and \(\vec a\cdot\vec b\) in terms of \(|\vec b|^2\).

Answer 1

Correct Answer: 60

\((\vec a+3\vec b)\cdot(7\vec a-5\vec b)=0\Rightarrow7|\vec a|^2+16\,\vec a\cdot\vec b-15|\vec b|^2=0\) ...(1)
\((\vec a-4\vec b)\cdot(7\vec a-2\vec b)=0\Rightarrow7|\vec a|^2-30\,\vec a\cdot\vec b+8|\vec b|^2=0\) ...(2)
(1) - (2): \(46\,\vec a\cdot\vec b-23|\vec b|^2=0\Rightarrow\vec a\cdot\vec b=\frac{|\vec b|^2}{2}\).
Put in (1): \(7|\vec a|^2+8|\vec b|^2-15|\vec b|^2=0\Rightarrow|\vec a|=|\vec b|\).
\(\cos\theta=\frac{\vec a\cdot\vec b}{|\vec a||\vec b|}=\frac12\Rightarrow\theta=60^\circ\).
Answer: 60