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

Intersection

To show that two straight lines intersect in 3D space, existence of a point which satisfies both lines must be proven.

It is not enough to show that the cross product of their parallel vectors is non-zero.

Normal to 2 lines

Let be two lines.

Here , are vectors parallel to respectively.

Normal to both lines: .

Unit normal

Angle between 2 straight lines

Using the lines mentioned above:

Here are vectors parallel to respectively.

Shortest distance from a point

The distance can be calculated using Pythogoras’ theorem.

Here:

• is the arbitrary point
• is the position vector of
• is the position vector of a point on the line
• is parallel to the line