Question
If $$\overrightarrow {AB} = \overrightarrow b $$ and $$\overrightarrow {AC} = \overrightarrow c $$ then the length of the perpendicular from $$A$$ to the line $$BC\,:$$
A.
$$\frac{{\left| {\overrightarrow b \times \overrightarrow c } \right|}}{{\left| {\overrightarrow b + \overrightarrow c } \right|}}$$
B.
$$\frac{{\left| {\overrightarrow b \times \overrightarrow c } \right|}}{{\left| {\overrightarrow b - \overrightarrow c } \right|}}$$
C.
$$\frac{1}{2}\frac{{\left| {\overrightarrow b \times \overrightarrow c } \right|}}{{\left| {\overrightarrow b - \overrightarrow c } \right|}}$$
D.
none of these
Answer :
$$\frac{{\left| {\overrightarrow b \times \overrightarrow c } \right|}}{{\left| {\overrightarrow b - \overrightarrow c } \right|}}$$
Solution :
$${\text{ar}}\left( {\Delta ABC} \right) = \frac{1}{2}\left| {\overrightarrow b \times \overrightarrow c } \right| = \frac{1}{2}\left| {\overrightarrow b - \overrightarrow c } \right|h,$$ where $$h = $$ the length of the perpendicular.
$$\therefore \,\,h = \frac{{\left| {\overrightarrow b \times \overrightarrow c } \right|}}{{\left| {\overrightarrow b - \overrightarrow c } \right|}}.$$