Vector Equations of Lines

This lesson covers the vector equation of a straight line, parallel lines, intersecting lines, and skew lines in 3D — as required by the Edexcel A-Level Mathematics specification (9MA0).

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The Vector Equation of a Line

A straight line in 3D can be described by giving:

• A point on the line (with position vector a)
• A direction vector d (parallel to the line)

The vector equation of the line is:

r = a + td

where:

• r is the position vector of a general point on the line
• a is the position vector of a known point on the line
• d is the direction vector of the line
• t is a scalar parameter (t can take any real value)

As t varies, r traces out every point on the line.

Example

Write the vector equation of the line through the point (2, 3, -1) with direction vector (1, -2, 4).

r = (2, 3, -1) + t(1, -2, 4)

In parametric form: x = 2 + t, y = 3 - 2t, z = -1 + 4t

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Finding the Equation from Two Points