Question

The distance from the point P(1,3,2) to the line through (1,0,0), and (1,2,0) is given by ?

Answer 1

Let M₁(1, 0, 0) and M₂(1, 2, 0).

We should find

$d=\frac{|\overrightarrow{PM_1}×\overrightarrow{s}|}{|\overrightarrow{s}|}$ , where $\overrightarrow{s}=\overrightarrow{M_1M_2}=\{ 1-1, 2-0, 0-0\}=\{ 0, 2, 0\}$ and $\overrightarrow{PM_1}=\{ 1-1, 0-3, 0-2\}=\{ 0, -3, -2\}$ .

$\overrightarrow{PM_1}×\overrightarrow{s}=\begin{vmatrix} \overrightarrow{i} & \overrightarrow{j}&\overrightarrow{k} \\ 0 & -3&-2\\ 0&2&0 \end{vmatrix}=\overrightarrow{i}(0+4)-\overrightarrow{j}(0-0)+\overrightarrow{k}(0-0)=4\overrightarrow{i}=\{ 4, 0, 0\}$ .

Then $d=\frac{\sqrt{4^2+0^2+0^2}}{\sqrt{0^2+2^2+0^2}}=\frac{4}{2}=2$ .