# Shortest distances

## Introduction to Shortest Distances

• In vector geometry, the shortest distance between two points is a straight line connecting them.
• This concept extends to finding the shortest distance between a point and a line, or between two lines.

## Shortest Distance from a Point to a Line

• To find the shortest distance from a point to a line, first express the line in parametric form: r = a + λb.
• If a is a point on the line and p is the given point, form the vector ap.
• Then, the vector ap can be split into two components: one parallel to the line (c) and one perpendicular to the line (which gives the shortest distance, d).
•  By forming the right-angled triangle apc, you can use Pythagoras’ Theorem to solve for d² = ap ² - c ².
• Alternatively, you can express the vector ap as a sum of the vector c and d, and then do a scalar (dot) product with b. As d.b = 0, you can isolate d.

## Shortest Distance between Two Lines

• To find the shortest distance between two skew (non-parallel, non-intersecting) lines, you need to construct a line perpendicular to both lines.
• Lines are given in Cartesian form as r = a + λb and r = p + µq, where a and p are points on the lines, and b and q are the directions of the lines.
• To construct this perpendicular line, you can use the cross product of b and q.
• This new line defines the common perpendicular between the original lines, and its length is the shortest distance between the lines.
•  Using the vector connecting any point on the first line to any point on the second line, ap (or pa), you can calculate the shortest distance as the scalar project of ap onto n (where **n = (b x q)/ b x q **).

## Shortest Distance between Two Objects in 3D Space

• The principles remain the same when working with points, lines, and planes in 3D space, but the calculations become a bit more complicated.
• When finding the shortest path between a point and a plane, you need to find the distance between the point and its perpendicular foot on the plane.
• When determining the shortest distance between two skew lines in 3D space, you need to find the length of the line segment connecting the two lines and may use line equations to do so.