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Grade 10 · Core Mathematics · Lesson 14 of 21

Determining Position Vectors and Calculating Vector Displacements

A Grade 10 Core Mathematics CBE lesson from the sub-strand Vectors I. In it, learners express 2D coordinates as column position vectors relative to the origin.

Strand Measurements and GeometrySub-strand Vectors ILesson code 10CM-28-L1440 minutes · 2 slides

KICD learning outcome (h)

Determine position vectors in different situations.

By the end of this lesson, learners should be able to

  1. Express 2D coordinates as column position vectors relative to the origin.
  2. Calculate the displacement vector AB⃗ given position vectors OA⃗ and OB⃗.
  3. Determine unknown coordinates of an initial or destination point given a displacement vector and one point.

Key vocabulary

Position vector
A vector that represents the location of a point relative to the origin.
The point K(4,−3) has a position vector OK⃗=(4−3).
Displacement vector
A vector indicating the change in position from a starting point to an endpoint.
The displacement vector AB⃗=(52) moves point A to point B.

Lesson preview

Slide 1 · Position Vectors and Relative DisplacementModel · 18 min
A coordinate plane showing points M and N, their position vectors from the origin, and the relative displacement vector MN.
Figure 1: A coordinate plane showing points M and N, their position vectors from the origin, and the relative displacement vector MN.

Any point on the Cartesian plane with coordinates (x,y) has a position vector OP⃗=(xy) measured from origin O(0,0). When finding the displacement vector AB⃗ between two points A and B, we subtract the position vector of the starting point from the position vector of the destination point, giving AB⃗=OB⃗−OA⃗. This algebraic rule lets us determine relative displacements or locate missing endpoint coordinates directly. Use Figure 1 to see how to calculate a relative displacement vector using position vectors.

  • A point P(x,y) has a unique position vector relative to the origin written as OP⃗=(xy).
  • The displacement vector connecting point A to point B is found using the formula AB⃗=OB⃗−OA⃗.
  • To calculate AB⃗, subtract the horizontal and vertical components of OA⃗ from those of OB⃗.
  • Given vector PQ⃗ and starting point P, destination point Q has position vector OQ⃗=OP⃗+PQ⃗.
Worked example. Write the position vectors using coordinates: OM⃗=(−43) and ON⃗=(5−2). Apply the displacement formula MN⃗=ON⃗−OM⃗. Subtract components: MN⃗=(5−2)−(−43)=(5−(−4)−2−3)=(9−5). Answer: OM⃗=(−43), ON⃗=(5−2), and MN⃗=(9−5)..

Check question: Point A has coordinates (3,−1) and AB⃗=(46). Find the position vector OB⃗ and the coordinates of point B.

  1. OB⃗=(75) and B(7,5)
  2. OB⃗=(−1−7) and B(−1,−7)
  3. OB⃗=(17) and B(1,7)
  4. OB⃗=(46) and B(4,6)
Show answer

OB⃗=(75) and B(7,5). Since AB⃗=OB⃗−OA⃗, we rearrange the formula to find OB⃗=OA⃗+AB⃗. Substituting the given vectors, we get OB⃗=(3−1)+(46)=(75). This gives point B(7,5).

Slide 2 · Solving Problems Involving Position Vectors and CoordinatesPractice · 22 min
A Cartesian grid displaying points E and F, illustrating their position vectors from the origin alongside the displacement vector EF.
Figure 2: A Cartesian grid displaying points E and F, illustrating their position vectors from the origin alongside the displacement vector EF.

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This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

  • Both slides with diagrams and class activities
  • Teacher notes, SNE adaptations and assessment checks
  • Homework and the other 20 lessons in this sub-strand
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All 21 lessons in Vectors I

  1. 1Explaining Vector and Scalar QuantitiesExplain vector and scalar quantities with illustrations
  2. 2Using Vector Notations in Different WaysUse vector notations in different ways
  3. 3Representing Vectors GeometricallyRepresent vectors geometrically in different situations
  4. 4Representing Vectors Geometrically in Different SituationsRepresent vectors geometrically in different situations
  5. 5Identifying Equivalent VectorsIdentify equivalent vectors in different situations
  6. 6Equivalent VectorsIdentify equivalent vectors in different situations
  7. 7Adding Vectors GeometricallyAdd vectors in different situations
  8. 8Adding Vectors Geometrically in Different SituationsAdd vectors in different situations
  9. 9Multiplying Vectors by ScalarsMultiply vectors by scalars
  10. 10Multiplying Vectors by Scalars: Graphical Construction and Scalar RatiosMultiply vectors by scalars
  11. 11Determining Column Vectors from Coordinates and DiagramsDetermine column vectors in different situations
  12. 12Determining Column Vectors and Vector ArithmeticDetermine column vectors in different situations
  13. 13Determining Position Vectors from the OriginDetermine position vectors in different situations
  14. 14Determining Position Vectors and Calculating Vector DisplacementsDetermine position vectors in different situations
  15. 15Calculating the Magnitude of a VectorWork out magnitude of a vector in different situations
  16. 16Working Out Vector Magnitudes in Different SituationsWork out magnitude of a vector in different situations
  17. 17Determining the Midpoint of a VectorDetermine the midpoint of a vector in different situations
  18. 18Determining Vector Midpoints and Endpoint CoordinatesDetermine the midpoint of a vector in different situations
  19. 19Determining Translation Vector as a TransformationDetermine translation vector as a transformation
  20. 20Determining Translation Vectors as TransformationsDetermine translation vector as a transformation
  21. 21Appreciating Vectors in Real-Life SituationsAppreciate the use of vectors in real-life situations