## 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.

_____________________________

d = √( (x1 - x2)2 + (y1 - y2)2 + (z1 - z2)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.