Vector Equation Of A Line

Hi there. In this math post, I cover the vector equation of a line.

Math images/text mostly from QuickLaTeX.com.

Vector Equation Of Line

Before giving the vector equation of a line, I cover the parts of the equation first.

We have a position vector of a known point on the line. This is in 3 dimensions.

There is a direction vector which tells how many units to move for x, y and z from the position vector.

A scalar parameter t is used that multiples the direction vector by any real number.

All of the above components gives the position vector r = (x, y, z) of any point on the line.

Note that negative values of the parameter t makes the direction vector go in the opposite direction.

Source1

Source 2: Nelson Calculus & Vectors Grade 12 Textbook for Ontario Math

Picture and example below from Jack's Math Youtube video.

Alternate Equation

The above vector equation can be also represented by:

This is also:

Example One

Find the vector equation of a line that passes through the point A(0, 2) with a direction vector of d = (4, 1).

We have the pieces for the vector equation formula. The position vector is the point A(0, 2). and the direction vector is d = (4, 1). Substitute accordingly.

I add the x co-ordinate pieces together to get 0 + 4t = 4t. Adding the y components gives 2 + t.

Example Two - Three Dimensions

Find the vector equation of a line passing through point A(1, 2, -3) with a direction vector d = (0, 2, -3).

This example is similar to the first example. We have three dimensions now.

Example 3

What if the direction vector is not given but you have two points? The two given points can be used to obtain a direction vector.

Find the equation of a line passing through C(3, -1, 7) and D(1, 2, 4).

The direction vector can be obtained by doing C minus D or D minus C. I do D minus C as I want to go from point C to point D.

Use point C as the position vector and the newly obtained direction vector for the vector equation formula.