Distance between point and line, Distance between point and plane

Coordinate Geometry (Analytic Geometry) in Three-dimensional Space

Point, Line and Plane - orthogonal projections, distances, perpendicularity of line and plane

Distance between point and line, example

Distance between point and plane, example

Distance between point and line

The distance d between a point and a line we calculate as the distance between the given point A(x₁, y₁, z₁) and its orthogonal projection onto the given line using the formula for the distance between two points.

The projection of the point onto the line is at the same time the intersection point A´(x_p, y_p, z_p) of the given line and a plane which passes through the given point orthogonal to the given line, thus

where d is the distance between the given point and the given line.

Example: Find the distance between a point A(-3, 5, -2) and a line

Solution: Through the given point lay a plane perpendicular to the given line, then as N = s = -i - j and

A(-3, 5, -2) => P :: -x - y + D = 0 gives -1 · (-3) - 1 · 5 + D = 0, D = 2 so, P :: -x - y + 2 = 0.

Then, rewrite the given line into the parametric form to get its intersection with the plane P,

plug these variable coordinates of a point of the line into the plane to find parameter t that determine the intersection point,

x = -t - 8 and y = -t => -x - y + 2 = 0 so that -(-t - 8) -(-t) + 2 = 0, t = -5

therefore, x = -(-5) - 8 = -3 and y = -t = -(-5) = 5, the intersection A´(-3, 5, 0).

The distance between the point A and the line equals the distance between points, A and A´ thus,

Distance between point and plane

We use the formulas for the distance from a point to a plane that are given in the two forms earlier in this chapter, they are

- the Hessian normal form, d = x₀cosa + y₀cosb + z₀cosg - p

- and when a plane is given in general form,

The distance between a point and a plane can also be calculated using the formula for the distance between two points, that is, the distance between the given point and its orthogonal projection onto the given plane.

Example: Given is a point A(4, 13, 11) and a plane x + 2y + 2z - 4 = 0, find the distance between the point and the plane.

Solution: Through the given point draw a line orthogonal to the plane, so that s = N = i + 2 j + 2k

The intersection of the line and the plane,

The distance between a point and the plane equals the distance between the point A and the intersection (or projection of the point A onto the plane), thus

Check the result using the above formula,

Coordinate geometry contents