Let for a triangle ABC, AB=-2i+j+3k CB= i+ j+ k CA=4i+3j+ k If 0 and the area of the tria…

Question

Let for a triangle $ABC$,

$\overset{	o \leftarrow }{AB}=-2\hat{i}+\hat{j}+3\hat{k}$

$\overset{	o \leftarrow }{CB}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$

$\overset{	o \leftarrow }{CA}=4\hat{i}+3\hat{j}+\delta \hat{k}$

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

Accepted Answer

$\begin{gathered} CA+AB=CB \ \Rightarrow +2i+4j+(\delta +3)k=\alpha i+\beta j+\gamma k \ \Rightarrow \alpha =+2,\beta =4,\gamma =\delta +3 \end{gathered}$

$\begin{gathered} =\frac{1}{2}|\overset{	o \leftarrow }{AB}	imes \overset{	o \leftarrow }{BC}|=|\frac{1}{2}|\begin{array}{ccc}\hat{i} \quad  \hat{j} \quad  \hat{k} \ -2 \quad  1 \quad  3 \ +2 \quad  4 \quad  \gamma \end{array}||=5\sqrt{6} \ =(\gamma -12{)}^{2}+(6+2\gamma {)}^{2}+100=(10\sqrt{6}{)}^{2} \ \Rightarrow 5{\gamma }^{2}=320 \end{gathered}$

$\begin{gathered} \Rightarrow {\gamma }^{2}=64 \ \ \Rightarrow \gamma =8 \ \ \Rightarrow \delta =5 \ \ \overset{	o \leftarrow }{CB}\cdot \overset{	o \leftarrow }{CA} =(2i+4j+8k)(4i+3j+5k) \ \ =8+12+40 \ \ =60 \end{gathered}$

Solutions · 1