Distance Between Point And Line

To find the perpendicular distance of the line from the point , we first express the line's parametric equations. The given symmetric form is:

From this, we derive the parametric equations:

The line's direction vector is and a point on the line is .

The vector from point to point on the line is:

The formula for the perpendicular distance from a point to a line given the direction vector and point vector is:

Calculating :

Now calculate its magnitude:

The perpendicular distance is:

The given line is in symmetric form. To find the perpendicular distance from the point to the line, we use the formula for the distance from a point to a line in space: where is the direction vector of the line, is the position vector of the point, and is a point on the line. The direction vector is , and . The vector . Now, applying the formula for distance: Thus, the perpendicular distance is .