Question

Let $a,b,c$ be unit vectors with $r=\alpha b\times c+\beta c\times a+\gamma a\times b,\left[abc\right]=2$ and the angle between $r$ and $a+b+c$ is $\frac{\pi }{4}$ with $\left|r\right|=\sqrt{2}$, then the maximum value of $\alpha +\beta +\gamma$ is

• A. $2\sqrt{2}$
• B. $6$
• C. $\frac{1}{\sqrt{2}}$
• D. $\frac{3}{2}$

Answer 1

The correct option is D $\frac{3}{2}$

Taking dot product with a on both the sides,we have
$r\cdot a=\alpha b\times c\cdot a=\alpha$ $[a$ $b$ $c]$
so $\alpha =\frac{1}{2}r\cdot a$. Similarly $\beta =\frac{1}{2}r\cdot b$ and $\gamma =\frac{1}{2}r\cdot c$
Hence $\alpha +\beta +\gamma =\frac{1}{2}r\cdot (a+b+c)$

$\le \frac{1}{2}\left|r\right||a+b+c|\cos \frac{\pi }{4}$

$\le \frac{1}{2}\sqrt{2}(\left|a\right|+\left|b\right|+\left|c\right|)\frac{1}{\sqrt{2}}=\frac{3}{2}$