Grade 10 · Core Mathematics · Lesson 10 of 21

Multiplying Vectors by Scalars: Graphical Construction and Scalar Ratios

A Grade 10 Core Mathematics CBE lesson from the sub-strand Vectors I. In it, learners construct scalar multiples of vectors and linear combinations on graph paper.

Strand Measurements and GeometrySub-strand Vectors ILesson code 10CM-28-L1040 minutes · 2 slides

KICD learning outcome (f)

Multiply vectors by scalars.

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

Construct scalar multiples of vectors and linear combinations on graph paper.
Determine the scalar factor k in a = kb by comparing non-zero vector components.

Key vocabulary

Linear combination: An expression formed by adding scalar multiples of two or more vectors together.; The vector $3 a + 2 b$ is a linear combination of vectors $a$ and $b$ .
Scalar factor: A real number multiplier that rescales the length of a vector without changing direction.; The scalar factor $k = 1.5$ increases the length of vector $g$ by half again.

Lesson preview

Slide 1 · Graphing Scalar Multiples and Vector CombinationsModel · 18 min

Figure 1: This diagram illustrates vectors a and b scaled on a grid, and then combined head-to-tail to form the resultant vector 3a + 2b.

To graph a scalar multiple of a vector on graph paper, we multiply its horizontal and vertical displacements by the scalar factor, keeping the direction unchanged if the scalar is positive (and reversing it if negative). When constructing a linear combination such as $3 a + 2 b$ , we first draw the scaled vector $3 a$ from an initial point. From its terminal point, we draw the second scaled vector $2 b$ head-to-tail. The resultant vector connects the starting point to the final head.

Multiplying a vector by a positive scalar $k$ increases its length by factor $k$ along the same direction.
To construct $3 a$ on grid paper, repeat the horizontal and vertical steps of $a$ three times continuously.
To draw $2 b$ , double both the horizontal and vertical displacements of vector $b$ .
To construct $3 a + 2 b$ , join vector $2 b$ to the tip of $3 a$ using head-to-tail addition.

Worked example. Calculate

3 a

: 3 times (2 units right, 1 unit up) gives 6 units right and 3 units up. Calculate

2 b

: 2 times (1 unit right, 3 units up) gives 2 units right and 6 units up. Add horizontal displacements: 6 units right + 2 units right = 8 units right. Add vertical displacements: 3 units up + 6 units up = 9 units up. Answer: The vector $3 a + 2 b$ goes 8 units right and 9 units up..

Check question: If vector $u$ moves 3 units right and 1 unit up, and vector $v$ moves 1 unit right and 2 units up, what is the horizontal displacement of $2 u + 3 v$ ?

4 units right
8 units right
9 units right
11 units right

Show answer

9 units right. Vector $2 u$ moves $2 \times 3 = 6$ units right. Vector $3 v$ moves $3 \times 1 = 3$ units right. The total horizontal displacement is $6 + 3 = 9$ units right.

Slide 2 · Fractional Scalar Multiples and Finding Scalar RatiosPractice · 22 min

Figure 2: This graph demonstrates the fractional scaling of vector g and compares parallel vectors a and b to determine their scalar ratio.

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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