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MathidiaMathematics Academy
Vectors in two and three dimensions
Lesson 41 of 75
Introduction

Vector equations of lines

Objective. Represent lines with position and direction vectors.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A vector equation of a line describes every point on it using a fixed position vector a and a direction vector b, written as r = a + tb, where t is a scalar parameter that varies over all real numbers. The point a lies on the line, and b gives the direction the line travels in three dimensions. For a line through (1,2,3) with direction (2,−1,0), the equation is r = (1,2,3) + t(2,−1,0). Changing t traces out every point on the line.

Worked example

Write the vector equation of the line passing through the points A(1,0,2) and B(3,4,−1).

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Illustrated card showing 1 … 0
  1. 1Find the direction vector AB = B − A = (2,4,−3)
  2. 2Choose position vector a = (1,0,2), a point on the line
  3. 3Form the equation r = a + t(AB)
  4. 4State r = (1,0,2) + t(2,4,−3)

Answer: r = (1,0,2) + t(2,4,−3)

Concept mastery

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Practice

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Common mistake. A common mistake is using two points directly as a and b instead of subtracting them to find the direction vector first. Always compute the direction as the difference between two points, not the points themselves.

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