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A2 Level · Pure Mathematics 4 WMA14

Vector Equations of Lines & The Scalar Product

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Vector Equations & Scalar Product

Edexcel International A Level Maths: Pure 4

What You'll Learn

Vector Equations of Lines — How to write the equation of a line using position and direction vectors
Line Relationships in 3D — Recognizing parallel, intersecting, and skew lines
The Scalar Product (Dot Product) — The two formulas and what they mean geometrically
Real Applications — Finding angles, testing perpendicularity, and calculating distances

Vector Equations of Lines

The Big Idea

A line can be described algebraically by giving two pieces of information:

One point on the line — described by its position vector a
The direction of the line — described by a direction vector d

Together, these create the vector equation of the line. Think of it like this: imagine you're standing at point a, and you want to walk in the direction of d. By walking different distances along that direction, you trace out the entire line.

The Two Main Formulas

Formula 1: When you know two points r = a + t(b − a)

What does this mean? If you have two points on the line (with position vectors a and b), the direction vector is simply b − a (the vector from the first point to the second). The parameter t is a scalar you can change; different values of t give you different points on the line.

Formula 2: When you know one point and a direction r = a + t d

What does this mean? If you have a point with position vector a and a known direction vector d, you can jump straight to this form. Again, t is the parameter that "travels" along the line.

Understanding the Parameter t

The parameter t is like a "distance dial." Each different value of t corresponds to a different point on the line:

When t = 0, you're at the point described by vector a
When t = 1, you're at the point a + d
When t = −1, you're at the point a − d (one step in the opposite direction)
Fractional and negative values of t are all allowed — they just put you at different locations along the (infinite) line

Worked Example 1

Find a vector equation of the line through points A(1, −8) and B(−3, 0).

Solution:

We have two points, so we use the formula r = a + t(b − a).

Position vector of A: a = (1, −8)

Position vector of B: b = (−3, 0)

Direction vector: b − a = (−3, 0) − (1, −8) = (−4, 8)

So the equation is:

r = (1, −8) + t(−4, 8)

Or equivalently (using a different point on the line):

r = (−3, 0) + t(−4, 8)

Or even (scaling the direction vector by −¼):

r = (−3, 0) + t(1, −2)

All three represent the same line! The direction vector can be any non-zero scalar multiple of the original.

Testing if a Point Lies on a Line

If you have the equation of a line and want to know whether a specific point P lies on it, substitute the coordinates of P into the equation and see if you can find a single value of t that works for all components.

Key Insight: single value of t

Why Many Equations Can Describe the Same Line

magnitude direction a

Example: These two equations describe the exact same line:

t = −2

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Also in the full note

Parallel, Intersecting & Skew Lines
The Scalar (Dot) Product
Uses of the Scalar Product
What to Memorise
Concepts Checklist
Exam Tips & Common Pitfalls
Why 3D is Different From 2D
Testing if Lines Are Parallel

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Vector Equations of Lines & The Scalar Product

Vector Equations & Scalar Product

What You'll Learn

Vector Equations of Lines

The Big Idea

The Two Main Formulas

Understanding the Parameter t

Worked Example 1

Testing if a Point Lies on a Line

Why Many Equations Can Describe the Same Line

More Pure Mathematics 4 topics