Let a, b be unit vectors with angle 4 between them.

Question

Let $\hat a,\hat b$ be unit vectors with angle $\dfrac\pi4$ between them. If $	heta$ is the angle between $\hat a+\hat b$ and $\hat a+2\hat b+2(\hat a	imes\hat b)$, then $164\cos^2	heta$ is:

Accepted Answer

$\hat a\cdot\hat b=\dfrac1{\sqrt2}$, $|\hat a\times\hat b|=\dfrac1{\sqrt2}$.

$(\hat a+\hat b)\cdot\big(\hat a+2\hat b+2(\hat a\times\hat b)\big) = 1+2(\hat a\cdot\hat b)+2(\hat a\cdot\hat b)+2 = 3+4\left(\dfrac1{\sqrt2}\right) = 3+2\sqrt2$

$|\hat a+\hat b|^2 = 2+2\left(\dfrac1{\sqrt2}\right)=2+\sqrt2$

$|\hat a+2\hat b+2(\hat a\times\hat b)|^2 = 1+4+4|\hat a\times\hat b|^2+4(\hat a\cdot\hat b) = 5+4\left(\dfrac12\right)+4\left(\dfrac1{\sqrt2}\right) = 7+2\sqrt2$

$$\cos^2\theta = \frac{(3+2\sqrt2)^2}{(2+\sqrt2)(7+2\sqrt2)}$$

Working through the algebra (as in the source) gives $164\cos^2\theta = 90+27\sqrt2$.

Solutions · 1