Question

$a,b,c$ are real. If $\theta$ is the angle between the vectors $ai+bj+ck$ and $bi+cj+ak$, then $\theta$ lies in the interval

• A. $(0,\frac{\pi }{2})$
• B. $(0,\frac{2\pi }{3})$
• C. $(0,\frac{\pi }{4})$
• D. $noneofthese$

Answer 1

The correct option is B $(0,\frac{2\pi }{3})$

We know that,

$\cos \theta =\frac{(ai+bj+ck)\cdot (bi+cj+ak)}{\sqrt{a^2+b^2+c^2}\sqrt{b^2+c^2+a^2}}$

where,

$\theta =$ angle between two vectors

$\therefore \cos \theta =\frac{ab+bc+ca}{a^2+b^2+c^2}$

Now, $(a+b+c{)}^2\ge 0$

$\therefore a^2+b^2+c^2\ge -2(ab+bc+ca)$

$\therefore -\frac{1}{2}\le \frac{ab+bc+ca}{a^2+b^2+c^2}$

$\therefore \theta \in (0,\frac{2pi}{3})$.