Question

We wish to find the distance from the point $P$to the line $L$as shown in the figure that follows. We know the coordinates of points $P$and $P0$but we do not know the coordinates of point $Q$

(a) If you knew the measure of angle $\theta$explain how you would find the distance from point $P$to line $L$

(b) Using a cross product, explain how you can find the distance from point $P$ to line $L$ even if you do not know the measure of angle $\theta$

Short answer

Part (a)$PQ=d\tan \theta$

Part (b)$\frac{||d\times \overrightarrow{P_{0}P}||}{\Vert d\Vert }$

Step-by-step solution

Part (a) Step 1: Given information

Consider a point $P$ to the line $\mathcal{L}$

Part (a) Step 2: Calculation

The goal is to calculate the distance between the points $P$and $Q$in the diagram.

The triangle $P{P}_{0}Q$is a right-angle triangle.

It can be written as, using trigonometric ratios.

$$\begin{gathered} \tan \theta =\frac{PQ}{P_{0}Q} \\ \tan \theta =\frac{PQ}{d} \end{gathered}$$

On both sides of the equation, multiply by $d$

$$\begin{gathered} d·\tan \theta =d·\frac{PQ}{d} \\ d·\tan \theta =PQ \end{gathered}$$

Thus, the distance $PQ=d\tan \theta$

Therefore, the answer is $PQ=d\tan \theta$

Part (b) Step 1: Calculation

The goal is to use the cross-product method to calculate the distance.

Now, suppose $Q$is the point on the line $\mathcal{L}$that is closest to the point $P$

From the figure, we can observe that,

$\Vert \overrightarrow{PQ}\Vert =||\overrightarrow{P_{0}P}||\sin \theta$, where $\theta$is the angle between $\overrightarrow{P_{0}P}$and the distance $d$

The distance between a point and a perpendicular line is defined by the theorem on perpendicular lines and points.

$$\begin{gathered} ||d\times \overrightarrow{P_{0}P}||=\Vert d\Vert \overrightarrow{P_{0}P}\Vert \sin \theta \\ \frac{||d\times \overrightarrow{P_{0}P}||}{\Vert d\Vert }=||\overrightarrow{P_{0}P}||\sin \theta \\ ||\overrightarrow{P_{0}P}||\sin \theta =\frac{||d\times \overrightarrow{P_{0}P}||}{\Vert d\Vert } \end{gathered}$$

Thus,

We know that $\Vert \overrightarrow{PQ}\Vert =||\overrightarrow{P_{0}P}||\sin \theta$

$\Vert \overrightarrow{PQ}\Vert =||\overrightarrow{P_{0}P}||\sin \theta =\frac{||d\times \overrightarrow{P_{0}P}||}{\Vert d\Vert }$

$\Vert \overrightarrow{PQ}\Vert =\frac{||d\times {\overrightarrow{P}}_{0}P||}{\Vert d\Vert }$

Therefore, the required distance using the cross product is $\frac{||d\times \overrightarrow{P_{0}P}||}{\Vert d\Vert }$