Shortest Distances

Question 1

How to find shortest distance between a point and a line?

Answer 1

1) Find vector between point and general point on the line, make this perpendicular to the direction vector of the line.
2) Use the dot product with this vector and the direction vector and equate them to 0.
3) Solve for lambda and substitute into vector between point and line.
4) Find magnitude of vector.

Question 2

How to find shortest distance between two parallel lines?

Answer 2

1) Find vector between a point on one line (position vector) and the general point on other line.
2) Equate the dot product of this vector and the direction vector to 0.
3) Solve for parameter (ų or lambda) and substitute back into vector between point and general point.
4) Find magnitude.

Question 3

How to find shortest distance between two skew lines?

Answer 3

1) Find vector between two general points of each line.
2) Dot this vector with both direction vectors, and equate to 0, to form simultaneous equations.
3) Solve equations and substitute back into vector between the two general points.
4) Find magnitude.