Grade 10 · Core Mathematics · Lesson 11 of 21

Determining Column Vectors from Coordinates and Diagrams

A Grade 10 Core Mathematics CBE lesson from the sub-strand Vectors I. In it, learners plot given points on a Cartesian plane and construct horizontal and vertical component lines.

Strand Measurements and GeometrySub-strand Vectors ILesson code 10CM-28-L1140 minutes · 2 slides

KICD learning outcome (g)

Determine column vectors in different situations.

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

Plot given points on a Cartesian plane and construct horizontal and vertical component lines.
Express vector displacements as column vectors using horizontal and vertical component additions.

Key vocabulary

Column vector: A pair of numbers stacked vertically representing horizontal and vertical displacements of a vector.; Vector $P Q$ is represented by the column vector $(\begin{array}{c} 4 \\ 3 \end{array})$ .
Displacement vector: A vector showing the straight-line change in position from a starting point to an ending point.; The displacement vector from $A (2, 1)$ to $B (5, 6)$ is vector $\vec{A B}$ .

Lesson preview

Slide 1 · Representing Displacements on a GridOpening · 18 min

Figure 1: A Cartesian grid displaying points P and Q, with horizontal and vertical lines intersecting at M to form a right-angled triangle.

When moving from an initial point to a terminal point on a grid, the total displacement combines horizontal and vertical movements. By drawing lines parallel to the horizontal and vertical axes through the endpoints, we form a right-angled triangle. The horizontal distance gives the $x$ -component and the vertical distance gives the $y$ -component of the column vector. These two perpendicular displacements combine to give the total displacement vector. Use Figure 1 to see how these horizontal and vertical lines form a right-angled triangle.

Plotting points on a Cartesian plane establishes precise starting and ending positions for a vector displacement.
A line drawn through the starting point parallel to the horizontal axis shows the $x$ -component.
A line drawn from the terminal point parallel to the vertical axis determines the $y$ -component.
The intersection of these grid lines forms a right-angled triangle that breaks the vector into components.

Worked example. Identify initial point

P (1, 2)

and terminal point

Q (5, 5)

. Find intersection point

M

of the horizontal line through

P

and vertical line through

Q

, giving

M (5, 2)

. Calculate horizontal component

\vec{P M} = 5 - 1 = 4

units to the right. Calculate vertical component

\vec{M Q} = 5 - 2 = 3

units upwards. Combine vector components to get

\vec{P Q} = \vec{P M} + \vec{M Q} = (\begin{array}{c} 4 \\ 3 \end{array})

. Answer: $\vec{P Q} = (\begin{array}{c} 4 \\ 3 \end{array})$ .

Check question: A vector starts at point $R (2, 3)$ and ends at point $S (6, 8)$ . If a horizontal line through $R$ intersects a vertical line through $S$ at point $N$ , what are the coordinates of $N$ and the column vector $\vec{R S}$ ?

$N (6, 3)$ and $\vec{R S} = (\begin{array}{c} 4 \\ 5 \end{array})$
$N (3, 6)$ and $\vec{R S} = (\begin{array}{c} 5 \\ 4 \end{array})$
$N (2, 8)$ and $\vec{R S} = (\begin{array}{c} 4 \\ 5 \end{array})$
$N (6, 3)$ and $\vec{R S} = (\begin{array}{c} 8 \\ 11 \end{array})$

Show answer

$N (6, 3)$ and $\vec{R S} = (\begin{array}{c} 4 \\ 5 \end{array})$ . Point $N$ shares the $x$ -coordinate of $S (6)$ and the $y$ -coordinate of $R (3)$ , so $N (6, 3)$ . The horizontal move is $6 - 2 = 4$ and vertical move is $8 - 3 = 5$ , giving $(\begin{array}{c} 4 \\ 5 \end{array})$ .

Slide 2 · Forming Column Vectors from Perpendicular ComponentsPractice · 22 min

Figure 2: A vector diagram showing horizontal vector CN and vertical vector ND being added together to find the resultant vector CD.

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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