Question
Distance of the point $$P\left( {\overrightarrow p } \right)$$ from the line $$\overrightarrow r = \overrightarrow a + \lambda \overrightarrow b $$ is :
A.
$$\left| {\left( {\overrightarrow a - \overrightarrow p } \right) + \frac{{\left( {\left( {\overrightarrow p - \overrightarrow a } \right).\overrightarrow b } \right)\overrightarrow b }}{{{{\left| {\overrightarrow b } \right|}^2}}}} \right|$$
B.
$$\left| {\left( {\overrightarrow b - \overrightarrow p } \right) + \frac{{\left( {\left( {\overrightarrow p - \overrightarrow a } \right).\overrightarrow b } \right)\overrightarrow b }}{{{{\left| {\overrightarrow b } \right|}^2}}}} \right|$$
C.
$$\left| {\left( {\overrightarrow a - \overrightarrow p } \right) + \frac{{\left( {\left( {\overrightarrow p - \overrightarrow b } \right).\overrightarrow b } \right)\overrightarrow b }}{{{{\left| {\overrightarrow b } \right|}^2}}}} \right|$$
D.
None of these
Answer :
$$\left| {\left( {\overrightarrow a - \overrightarrow p } \right) + \frac{{\left( {\left( {\overrightarrow p - \overrightarrow b } \right).\overrightarrow b } \right)\overrightarrow b }}{{{{\left| {\overrightarrow b } \right|}^2}}}} \right|$$
Solution :
Let $$Q\left( {\overrightarrow q } \right)$$ be the foot of altitude drawn from $$P\left( {\overrightarrow p } \right)$$ to the line $$\overrightarrow r = \overrightarrow a + \lambda \overrightarrow b ,$$
$$\eqalign{
& \Rightarrow \left( {\overrightarrow q - \overrightarrow p } \right).\overrightarrow b = 0{\text{ and }}\overrightarrow q = \overrightarrow a + \lambda \overrightarrow b \cr
& \Rightarrow \left( {\overrightarrow a + \lambda \overrightarrow b - \overrightarrow p } \right).\overrightarrow b = 0 \cr
& {\text{or }}\left| {\left( {\overrightarrow a - \overrightarrow p } \right)} \right|.\overrightarrow b + \lambda {\left| {\overrightarrow b } \right|^2} = 0\,\,{\text{or }}\lambda = \frac{{\left( {\overrightarrow p - \overrightarrow a } \right).\overrightarrow b }}{{{{\left| {\overrightarrow b } \right|}^2}}} \cr
& \Rightarrow \overrightarrow q - \overrightarrow p = \overrightarrow a + \frac{{\left( {\left( {\overrightarrow p - \overrightarrow a } \right).\overrightarrow b } \right)\overrightarrow b }}{{{{\left| {\overrightarrow b } \right|}^2}}} - \overrightarrow p \cr
& \Rightarrow \left| {\overrightarrow q - \overrightarrow p } \right| = \left| {\left( {\overrightarrow a - \overrightarrow p } \right) + \frac{{\left( {\left( {\overrightarrow p - \overrightarrow a } \right).\overrightarrow b } \right)\overrightarrow b }}{{{{\left| {\overrightarrow b } \right|}^2}}}} \right| \cr} $$