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Grade 10 · Core Mathematics · Lesson 12 of 21

Determining Column Vectors and Vector Arithmetic

A Grade 10 Core Mathematics CBE lesson from the sub-strand Vectors I. In it, learners determine the column vector between two given coordinate points algebraically and geometrically.

Strand Measurements and GeometrySub-strand Vectors ILesson code 10CM-28-L1240 minutes · 2 slides

KICD learning outcome (g)

Determine column vectors in different situations.

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

  1. Determine the column vector between two given coordinate points algebraically and geometrically.
  2. Calculate scalar multiples and linear combinations of column vectors.

Key vocabulary

Column vector
A two-row column matrix displaying the horizontal and vertical displacements of a vector.
The column vector (58) shows 5 units right and 8 units up.
Horizontal component
The distance and direction moved along or parallel to the x-axis in coordinate geometry.
In vector CD⃗=(6−3), the horizontal component is positive 6.
Vertical component
The distance and direction moved along or parallel to the y-axis in coordinate geometry.
In column vector (4−7), the vertical component is negative 7.
Linear combination
An expression formed by multiplying vectors by scalars and adding or subtracting the results.
The vector 2u⃗+3v⃗ is a linear combination of column vectors u⃗ and v⃗.

Lesson preview

Slide 1 · Determining Column Vectors from CoordinatesModel · 18 min
A coordinate grid displaying the vector KL alongside its horizontal and vertical displacements from point K to point L.
Figure 1: A coordinate grid displaying the vector KL alongside its horizontal and vertical displacements from point K to point L.

A vector displacement describes movement from an initial point to a terminal point. We express any displacement using horizontal and vertical components together, combined into a single column vector (xy). When given coordinate points A(x1,y1) and B(x2,y2), the column vector AB⃗ is found algebraically by subtracting initial coordinates from terminal coordinates, giving (x2−x1y2−y1).

  • A column vector records horizontal displacement in the top entry and vertical displacement in the bottom entry.
  • Rightward and upward displacements are positive, whereas leftward and downward displacements are written as negative values.
  • To determine vector AB⃗ algebraically, subtract coordinates of starting point A from ending point B.
  • Learners discuss their component count against coordinate subtraction to confirm both methods produce identical results.
Worked example. Identify initial point K(−2,5) as (x1,y1) and terminal point L(4,−1) as (x2,y2). Calculate horizontal component x=x2−x1=4−(−2)=4+2=6. Calculate vertical component y=y2−y1=−1−5=−6. Combine components into column vector format KL⃗=(6−6). Answer: KL⃗=(6−6).

Check question: Points R and S have coordinates R(3,−2) and S(−1,4). What is the column vector RS⃗?

  1. (−46)
  2. (4−6)
  3. (22)
  4. (−42)
Show answer

(−46). Subtracting coordinates of initial point R from terminal point S yields x=−1−3=−4 and y=4−(−2)=6, giving column vector (−46).

Slide 2 · Scalar Multiplication and Linear CombinationsPractice · 22 min
A coordinate grid illustrating the geometric addition of two scaled vectors to form a resultant column vector.
Figure 2: A coordinate grid illustrating the geometric addition of two scaled vectors to form a resultant column vector.

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