Mathematics - Vector Question with Solution | TestHub

$$\begin{array}{rlrlr}& \alpha =(\overline{\mathrm{AD}},\overline{\mathrm{BC}})={\cos }^{-1}\frac{1}{4}\Rightarrow \cos \alpha =\frac{1}{4}=\frac{\overline{\mathrm{AD}}\cdot \overline{\mathrm{BC}}}{|\overline{\mathrm{AD}}||\overline{\mathrm{BC}}|}=\frac{\overline{\mathrm{d}}\cdot (\overline{\mathrm{c}}-\overline{\mathrm{b}})}{|\overline{\mathrm{d}}||\overline{\mathrm{c}}-\overline{\mathrm{b}}|}& & & \end{array}$$

$$=4\overline{\mathrm{d}}\cdot (\overline{\mathrm{c}}-\overline{\mathrm{b}})=|\overline{\mathrm{d}}||\overline{\mathrm{c}}-\overline{\mathrm{b}}|=|\overline{\mathrm{b}}||\overline{\mathrm{d}}|$$

$$Since|\overline{\mathrm{b}}-\overline{\mathrm{c}}|=|\overline{\mathrm{b}}|=|\overline{\mathrm{c}}|,Let(\overline{\mathrm{d}},\overline{\mathrm{b}})=\theta ,(\overline{\mathrm{d}},\overline{\mathrm{c}})=\frac{\pi }{3}$$

$$\Rightarrow 4\left(|\overline{\mathrm{d}}||\overline{\mathrm{c}}|\frac{1}{2}-|\overline{\mathrm{d}}||\overline{\mathrm{b}}|\cos \theta \right)=|\overline{\mathrm{d}}||\overline{\mathrm{b}}|$$

$$\begin{array}{rlrlrlrlr}& \Rightarrow 4\left(\frac{1}{2}-\cos \theta \right)=1\Rightarrow \cos \theta =\frac{1}{4}& & & & & & & \end{array}$$

$$\overline{\mathrm{ABC}}{\perp }^{\ln }\overline{\mathrm{DBC}}\Rightarrow (\overline{\mathrm{AB}}\times \overline{\mathrm{AC}})\cdot (\overline{\mathrm{BD}}\times \overline{\mathrm{DC}})=0$$

$$\Rightarrow (\overline{\mathrm{b}}\times \overline{\mathrm{c}})\cdot ((\overline{\mathrm{b}}-\overline{\mathrm{d}})\times (\overline{\mathrm{c}}-\overline{\mathrm{d}}))=0\Rightarrow (\overline{\mathrm{b}}\times \overline{\mathrm{c}})\cdot (\overline{\mathrm{b}}\times \overline{\mathrm{c}}-\overline{\mathrm{b}}\times \overline{\mathrm{d}}-\overline{\mathrm{d}}\times \overline{\mathrm{c}})=0$$

$$\Rightarrow |\overline{\mathrm{b}}\times \overline{\mathrm{c}}|-(\overline{\mathrm{b}}\times \overline{\mathrm{c}})\cdot (\overline{\mathrm{b}}\times \overline{\mathrm{d}})-(\overline{\mathrm{b}}\times \overline{\mathrm{c}})\cdot (\overline{\mathrm{d}}\times \overline{\mathrm{c}})=0$$