Question

The points $O,A,B,C,D$ are such that $OA=a,OB=b,OC=2a+3b$ and $OD=a-2b.$ If $\left|a\right|=3\left|b\right|,$ then the angle between $BD$ and $AC$ is

• A. $\pi$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{3}$
• D. None of these

Answer 1

Answer: (B)

$\frac{\pi }{2}$

$BD.AC=\left|BD\right|\left|AC\right|\cos \theta$ where $\theta$ is the angle between $BD$ and $AC.$

$\Rightarrow (OD-OB).(OC-OA)=|OD-OB||OC-OA|\cos \theta \Rightarrow (a-2b-b).(2a+3b-a)=|a-3b||a+3b|\cos \theta \Rightarrow a^2-9b^2=|a-3b||a+3b|\cos \theta \Rightarrow 0=|a-3b||a+3b|\cos \theta [\because \left|a\right|=3\left|b\right|,\therefore a^2=9b^2]$

$\Rightarrow \cos \theta =0\Rightarrow \theta =\frac{\pi }{2}$