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Grade 10 · Core Mathematics · Lesson 13 of 21

Determining Position Vectors from the Origin

A Grade 10 Core Mathematics CBE lesson from the sub-strand Vectors I. In it, learners express coordinates of points on a Cartesian plane as column position vectors from the origin.

Strand Measurements and GeometrySub-strand Vectors ILesson code 10CM-28-L1340 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 coordinates of points on a Cartesian plane as column position vectors from the origin.
  2. Calculate displacement vectors between two points using their position vectors.

Key vocabulary

Position vector
A vector that specifies the position of a point relative to a fixed origin.
Point A(3,5) has position vector OA⃗=(35).
Origin
The fixed reference point (0,0) on a Cartesian plane where axes intersect.
All position vectors are measured starting from the origin O(0,0) on the graph grid.
Column vector
A vector written with its components stacked vertically inside brackets.
The position of a point is written as a column vector (xy).

Lesson preview

Slide 1 · Connecting Coordinates to Position VectorsModel · 18 min
A Cartesian plane displaying three position vectors drawn from the origin to specific points located in different quadrants.
Figure 1: A Cartesian plane displaying three position vectors drawn from the origin to specific points located in different quadrants.

On a Cartesian plane, every point has a location defined by its ordered pair (x,y). When we draw a directed line segment from the fixed origin O(0,0) to a point P(x,y), we create a position vector OP⃗. The x-coordinate gives the horizontal movement, and the y-coordinate gives the vertical movement. Thus, the position vector corresponds directly to its coordinates expressed as a column vector (xy). Use Figure 1 to observe how a position vector relates to its coordinates.

  • The origin O(0,0) acts as the common starting reference point for all position vectors.
  • A point with coordinates (x,y) has position vector OP⃗=(xy) from the origin.
  • The upper number in a column vector represents horizontal movement from the origin.
  • The lower number in a column vector represents vertical movement from the origin.
Worked example. 1. Identify the x and y coordinates for point A(5,8), which gives OA⃗=(58). 2. Identify the x and y coordinates for point B(−4,3), which gives OB⃗=(−43). 3. Identify the x and y coordinates for point C(−2,−6), which gives OC⃗=(−2−6). Answer: OA⃗=(58), OB⃗=(−43), OC⃗=(−2−6).

Check question: What is the position vector of point W(−6,9) relative to the origin O?

  1. (−69)
  2. (9−6)
  3. (−6,9)
  4. (6−9)
Show answer

(−69). The x-coordinate is −6 and the y-coordinate is 9. A position vector is written vertically as a column vector (xy), so placing the components in this format gives (−69).

Slide 2 · Determining Vectors Between Points using Position VectorsPractice · 22 min
A Cartesian plane illustrating the position vectors of two points and the displacement vector connecting them.
Figure 2: A Cartesian plane illustrating the position vectors of two points and the displacement vector connecting them.

Teach the full lesson on the projector

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