What angle do the unit vectors vec m and vec n make if the…

What angle do the unit vectors $\vec m$ and $\vec n$ make if the vectors $\vec p=\vec m+2\vec n$ and $\vec q=5\vec m-4\vec n$ are perpendicular?

Explanation of Method

The angle between unit vectors whose linear combinations are perpendicular.

Step 1: Perpendicularity

\[ \vec p\cdot\vec q=(\vec m+2\vec n)\cdot(5\vec m-4\vec n)=5\vec m^{\,2}-4\,\vec m\cdot\vec n+10\,\vec n\cdot\vec m-8\vec n^{\,2}=5+6\,\vec m\cdot\vec n-8=0. \]

Step 2: The angle

\[ \vec m\cdot\vec n=\tfrac12=\cos\varphi\quad(|\vec m|=|\vec n|=1). \]

\[\varphi=60^\circ\]

Check: $5+6\cdot\tfrac12-8=0$ $\checkmark$

$60^\circ$ $O$ $\vec m$ $\vec n$ $\vec p$ $\vec q$

$\vec p=\vec m+2\vec n$
$\vec q=5\vec m-4\vec n$

With $\angle(\vec m,\vec n)=60^\circ$ the vectors $\vec p$ and $\vec q$ are perpendicular (dotted: the parallelogram constructions).

Result

\[\boxed{\,\varphi=60^\circ\,}\]