Calculate distances involving points, lines, and planes

Distance is understood as the minimum Euclidean length between geometric objects: point-to-point distance, the perpendicular distance from a point to a line or plane, and the separation of parallel lines or planes. Using coordinates, vector representations, and equations, a learner determines when intersecting objects have distance zero and identifies the common perpendicular for parallel or skew lines; the scope excludes higher-dimensional metric spaces and more advanced abstract or optimization-based treatments.

Detailed Explanation: Calculate distances involving points, lines, and planes

Distance means the shortest Euclidean length between two objects. For a point and a plane, the shortest segment is always perpendicular to the plane.

For a point $(P(x_0,y_0,z_0))$ and a plane

Ax+By+Cz+D=0,

the perpendicular distance is

d=\frac{ \vert Ax_0+By_0+Cz_0+D \vert }{\sqrt{A^2+B^2+C^2}}.

The vector

\mathbf n=\langle A,B,C\rangle

is a normal vector to the plane, so it gives the direction of the shortest path.

Worked example

Find the distance from

P(1,2,3)

to the plane

2x-y+2z-3=0.

Step 1: Identify the coefficients

Compare the plane with $(Ax+By+Cz+D=0)$ :

A=2,\qquad B=-1,\qquad C=2,\qquad D=-3.

Step 2: Substitute the point’s coordinates

Calculate the numerator:

\vert Ax_0+By_0+Cz_0+D \vert = \vert 2(1)-1(2)+2(3)-3 \vert .

= \vert 2-2+6-3 \vert =3.

Step 3: Calculate the length of the normal vector

\sqrt{A^2+B^2+C^2} = \sqrt{2^2+(-1)^2+2^2} = \sqrt{9}=3.

Step 4: Divide

d=\frac{3}{3}=1.

Therefore, the distance from $P$ to the plane is

\boxed{1\text{ unit}}.

The shortest segment from $P$ to the plane follows the normal direction $\langle 2,-1,2\rangle$ . If a point already lies on the plane, substituting its coordinates gives $0$ , so its distance to the plane is $0$ . Similarly, intersecting lines or planes have distance $0$ ; only separate parallel or skew objects have a positive distance.

Learn by doing: Calculate distances involving points, lines, and planes

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3D Space - Plane and Line - Scalar and Vector Equations to Closest Distance

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