If a, b are unit vectors, the greatest value of √3| a+ b|+| a- b| is _______.

Question

If $\vec a,\vec b$ are unit vectors, the greatest value of $\sqrt3|\vec a+\vec b|+|\vec a-\vec b|$ is \_\_\_\_\_\_\_.

Answer: 4

Accepted Answer

With $\theta$ the angle between $\vec a,\vec b$: $|\vec a+\vec b|=2\cos\dfrac\theta2$, $|\vec a-\vec b|=2\sin\dfrac\theta2$.

$$\sqrt3|\vec a+\vec b|+|\vec a-\vec b| = 2\sqrt3\cos\frac\theta2+2\sin\frac\theta2$$

Maximum value $=\sqrt{(2\sqrt3)^2+2^2} = \sqrt{16}=4$.

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