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  • Let for a triangle ABC, <img alt="\begin{aligned} & \overrightarrow{\mathrm{AB}}=-2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}} \\ & \overrightarrow{\mathrm{CB}}=\alpha \

Let for a triangle ABC,

\begin{aligned} & \overrightarrow{\mathrm{AB}}=-2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}} \\ & \overrightarrow{\mathrm{CB}}=\alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}} \\ & \overrightarrow{\mathrm{CA}}=4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\delta \hat{\mathrm{k}} \end{aligned}

If  \delta>0 and the area of the triangle  \mathrm{ABC}  is  5 \sqrt{6} , then \overrightarrow{\mathrm{CB}} \cdot \overrightarrow{\mathrm{CA}}  is equal to

Option: 1

108


Option: 2

60


Option: 3

54

 


Option: 4

120


Answers (1)

best_answer

\begin{aligned} & \overrightarrow{\mathrm{CA}}+\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{CB}} \\ & \langle 4,3, \delta\rangle \cdot+\langle-2,1,3\rangle=\overrightarrow{\mathrm{CB}} \\ & \Rightarrow \overrightarrow{\mathrm{CB}}=\langle 2,4,3+\delta\rangle \end{aligned}

\begin{aligned} & \overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{AC}}=\left|\begin{array}{ccc} \hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ -2 & 1 & 3 \\ -4 & -3 & -\delta \end{array}\right| \\ & =\hat{\ell}(9-\delta)-\hat{\mathrm{j}}(2 \delta+12)+\mathrm{k}(10) \\ & |\mathrm{AB} \times \mathrm{AC}|^2=(9-\delta)^2+(2 \delta+12)^2+(10)^2 \\ & =5 \delta^2+30 \delta+325 \\ & \text { Area of } \Delta \mathrm{ABC}=5 \sqrt{6} \\ & \Rightarrow \frac{1}{2}|\overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{AC}}|=5 \sqrt{6} \end{aligned}

\begin{aligned} & \Rightarrow|\overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{AC}}|^2=600 \\ & \Rightarrow 5 \delta^2+30 \delta-275=0 \\ & \Rightarrow \mathrm{S}^2+6 \delta-55=0 \\ & \Rightarrow(\delta+11)(\delta-5)=0 \\ & \delta=5 \\ & \overrightarrow{\mathrm{CB}}=<2,3,8> \\ & \overrightarrow{\mathrm{CB}} \cdot \overrightarrow{\mathrm{CA}} \cdot=<2,4,8>\cdot<4,3,5> \\ & =8+12+40=60 \end{aligned}

 

Posted by

Irshad Anwar

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