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MathidiaMathematics Academy
Vectors and geometric proof
Lesson 47 of 75
Easy practice

Position vectors

Objective. Express points and displacements relative to an origin.

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

Learn: the key idea

Position vectors describe the location of a point relative to a fixed origin O, written as the vector from O to that point, such as OA for point A. The displacement vector between two points with known position vectors is found by subtracting: AB = OB − OA. If A has position vector (2, 3) and B has position vector (7, 1), then AB = (7−2, 1−3) = (5, −2).

Worked example

Points A and B have position vectors OA = (1, 5) and OB = (6, −2). Find the vector AB in terms of position vectors.

1 … 5
Illustrated card showing 1 … 5
  1. 1Recall AB = OB − OA
  2. 2Substitute the position vectors: AB = (6, −2) − (1, 5)
  3. 3Subtract component-wise: AB = (6−1, −2−5)
  4. 4Simplify: AB = (5, −7)

Answer: (5, −7)

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Practice

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Common mistake. A common mistake is calculating AB as OA − OB instead of OB − OA, which reverses the direction of the resulting vector. Always subtract the starting point's position vector from the ending point's position vector.

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