Let a, b, c be mutually perpendicular vectors of equal magnitude, equally inclined at angl…

Question

Let $\vec a,\vec b,\vec c$ be mutually perpendicular vectors of equal magnitude, equally inclined at angle $	heta$ to $\vec a+\vec b+\vec c$. Then $36\cos^2(2	heta)$ is \_\_\_\_\_\_\_.

Answer: 4

Accepted Answer

$$\vec a\cdot(\vec a+\vec b+\vec c) = |\vec a|^2 = |\vec a|\cdot\sqrt3|\vec a|\cos\theta \implies 1=\sqrt3\cos\theta \implies \cos\theta=\frac1{\sqrt3}$$

$$\cos2\theta = 2\cos^2\theta-1 = \frac23-1=-\frac13 \implies 36\cos^2(2\theta) = 36\cdot\frac19=4$$

Solutions · 1