## Distances between Points, Lines and Planes.

### Distance between two Points:

Given A(2|1|-7), B(5|5|5) Distance between A and B : ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ d(A,B) = 13

It's calculated by the Pythagorean formula.

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d = √( (x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2} + (z_{1} - z_{2})^{2})

### Distance between a point and a line:

Given P(2|0|3) −> ⎧ 1 ⎫ ⎧ 1 ⎫ g : x = ⎪ 1 ⎪ + s·⎪ 0 ⎪ ⎩ 0 ⎭ ⎩-2 ⎭ Distance between P and g: ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ d(P,g) = 2,4494897

Take the plane E in normal form with the point P as position vector and the direction of the line g as normal vector. Determine the point of intersection point S between this plane and the line g. The distance between S and P is the distance between the point and the line.

### Distance between Point and Plane:

Given P(0|0|0) E : x + y = 1 Distance between P and E : ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ d(P,E) = -0,70710678

Intersect the plane by the perpendicular from the point to the plane and determine the distance between the point of intersection and the given point.

### Distance between two Lines:

### Distance between Plane and Line:

See Intersection of Plane and Line