If a, b, c are unit vectors with a+2 b+2 c= 0, then | a c| is:

Question

If $\vec a,\vec b,\vec c$ are unit vectors with $\vec a+2\vec b+2\vec c=\vec 0$, then $|\vec a	imes\vec c|$ is:

Accepted Answer

$\vec a=-2\vec b-2\vec c$: $|\vec a|^2=4+4+8\vec b\cdot\vec c=1 \Rightarrow \vec b\cdot\vec c=-\dfrac78$.

$\vec a\cdot\vec b = -2-2\vec b\cdot\vec c = -2+\dfrac74=-\dfrac14$; similarly $\vec a\cdot\vec c=-\dfrac14$.

$$|\vec a\times\vec c|^2 = |\vec a|^2|\vec c|^2-(\vec a\cdot\vec c)^2 = 1-\frac1{16}=\frac{15}{16} \implies |\vec a\times\vec c|=\frac{\sqrt{15}}4$$

Solutions · 1