Vector Equation of Straight Lines

Passes through a point & parallel to a vector

Equation for a line that:

• passes through
• is parallel to

Parametric equation

Symmetric equation

Passes through 2 points

Equation of a line passes through , . and are the position vectors of and .

Parametric equation

Symmetric equation

Note

To show that two straight lines intersect in 3D space, it is not enough to show that the cross product of their parallel vectors is non-zero.

Also: Existence of a point which satisfies both lines must be proven.

Normal to 2 lines

Let be two lines.

Here , are vectors parallel to respectively.

Normal to both lines: . Unit normal to both lines can be found by:

Angle between 2 straight lines

Using the lines mentioned above:

Here are vectors parallel to respectively.

Shortest distance to a point

Suppose and lie on a line. Shortest distance to the point is: