Practicing Success
If $\vec a,\vec b$ are unit vectors such that $|\vec a+\vec b|=1$ and $|\vec a-\vec b|=\sqrt{3}$, then $|3\vec a+2\vec b|=$ |
7 4 $\sqrt{7}$ $\sqrt{19}$ |
$\sqrt{7}$ |
Let θ be the angle between a and b. Then, $\tan\frac{θ}{2}=\frac{|\vec a-\vec b|}{|\vec a+\vec b|}⇒\tan\frac{θ}{2}=\sqrt{3}⇒θ=120°$ $∴\vec a.\vec b=|\vec a||\vec b|\cos θ=\cos 120°=-\frac{1}{2}$ Now, $|3\vec a+2\vec b|^2=9|\vec a|^2+4|\vec b|^2+12(\vec a.\vec b)$ $⇒|3\vec a+2\vec b|^2=9+4+12×(-\frac{1}{2})=7$ $⇒|3\vec a+2\vec b|=\sqrt{7}$ |