Let in a A B C, the length of the side A C be 6 , the vertex B be (1,2,3) and the vertices…

Question

Let in a $	riangle A B C$, the length of the side $A C$ be 6 , the vertex $B$ be $(1,2,3)$ and the vertices $A, C$ lie on the line $\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$. Then the area (in sq. units) of $	riangle A B C$ is:

Accepted Answer

$$\begin{aligned}
& \text { Let } \mathrm{M}(3 \lambda+6,2 \lambda+7,-2 \lambda+7) \
& \overrightarrow{\mathrm{BM}}=(3 \lambda+5) \hat{\mathrm{i}}+(2 \lambda+5) \hat{\mathrm{j}}+(-2 \lambda+4) \hat{\mathrm{k}} \
& \overrightarrow{\mathrm{AC}} \cdot \overrightarrow{\mathrm{BM}}=0=3(3 \lambda+5)+2(2 \lambda+5)-2(-2 \lambda+4) \
& \overrightarrow{\mathrm{BM}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+6 \hat{\mathrm{k}} \
& \mid \overrightarrow{\mathrm{BM}}=7 \
& \text { Area }=\frac{1}{2} \times 6 \times 7=21
\end{aligned}$$

Solutions · 1