Q .55. - Calculus

Question

. Prove that the distance from the point P to the line given by the equation 

$\mathbf{r}\left(t\right)={\mathbf{P}}_0+t\mathbf{d}\text{ is given by }\frac{||\mathbf{d}\times P_{0}P||}{\Vert \mathbf{d}\Vert }$

Step-by-Step Solution

Answer

.

Step 1: Set up the geometry

The line is \( \mathbf{r}(t) = \mathbf{P}_0 + t\mathbf{d} \). The vector from \( \mathbf{P}_0 \) to \( P \) is \( \overrightarrow{P_0P} \).

Step 2: Project and find distance

The distance from \( P \) to the line is the component of \( \overrightarrow{P_0P} \) perpendicular to \( \mathbf{d} \). By the cross product formula, this equals \( \frac{|\mathbf{d} \times \overrightarrow{P_0P}|}{|\mathbf{d}|} \). \( \blacksquare \)