A vector m has a magnitude of 5 units pointing on a bearing of 045°. What are the magnitude and direction of the vector 2.5m?

Magnitude 12.5 units, bearing of 045°. Multiplying vector m by the positive scalar 2.5 increases its magnitude to 2.5 × 5 = 12.5 units. Multiplying by a positive scalar leaves the original direction unchanged, so the bearing remains 045°.

Grade 10 · Core Mathematics · Lesson 9 of 21

Multiplying Vectors by Scalars

A Grade 10 Core Mathematics CBE lesson from the sub-strand Vectors I. In it, learners multiply a vector quantity by a positive scalar geometrically on grid paper.

Strand Measurements and GeometrySub-strand Vectors ILesson code 10CM-28-L940 minutes · 2 slides

KICD learning outcome (f)

Multiply vectors by scalars.

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

Multiply a vector quantity by a positive scalar geometrically on grid paper.
Construct resultant vectors formed by linear combinations of scaled vectors.

Key vocabulary

Scalar multiplication: The operation of multiplying a vector by a real number scalar quantity.; In scalar multiplication, multiplying vector $p$ by $3$ gives the vector $3 p$ .
Resultant vector: The single vector obtained by combining two or more vectors together.; The resultant vector of $p + p + p$ is equal to $3 p$ .
Linear combination: An expression formed by multiplying vectors by scalars and adding the results.; The vector expression $3 a + 2 b$ is a linear combination of $a$ and $b$ .

Lesson preview

Slide 1 · Multiplying a Vector by a Positive ScalarModel · 18 min

Figure 1: A vector p is multiplied by a scalar of three, showing three identical vectors placed head-to-tail to form the resultant vector 3p.

When a vector quantity is multiplied by a positive scalar $k$ , its magnitude is scaled by the factor $k$ , while its direction remains unchanged. Adding a vector to itself repeatedly, such as $p + p + p$ , gives the resultant vector $3 p$ . The length of $3 p$ is three times the magnitude of $p$ , pointing in the exact same direction as $p$ . Use Figure 1 to see how adding a vector to itself scales its magnitude.

A vector quantity can be multiplied by a scalar factor to change its magnitude.
Multiplying a vector by a positive scalar preserves the original direction of the vector.
Repeatedly adding a vector to itself $n$ times produces the scalar multiple $n v$ .
The magnitude of $k v$ is equal to $k$ times the magnitude of $v$ .

Worked example. Identify the magnitude of

u

, which is

4 cm

, and its direction, which is East. Multiply the magnitude by the scalar factor

3

∣ 3 u ∣ = 3 \times 4 cm = 12 cm

. Keep the direction the same as

u

, which is East. Draw a directed line segment of length

12 cm

pointing East. Answer: The magnitude of $3 u$ is $12 cm$ and its direction is due East..

Check question: A vector $m$ has a magnitude of $5 units$ pointing on a bearing of $045^{\circ}$ . What are the magnitude and direction of the vector $2.5 m$ ?

Magnitude $12.5 units$ , bearing of $045^{\circ}$
Magnitude $12.5 units$ , bearing of ${112.5}^{\circ}$
Magnitude $7.5 units$ , bearing of $045^{\circ}$
Magnitude $5 units$ , bearing of ${112.5}^{\circ}$

Show answer

Magnitude $12.5 units$ , bearing of $045^{\circ}$ . Multiplying vector $m$ by the positive scalar $2.5$ increases its magnitude to $2.5 \times 5 = 12.5 units$ . Multiplying by a positive scalar leaves the original direction unchanged, so the bearing remains $045^{\circ}$ .

Slide 2 · Constructing Linear Combinations of VectorsPractice · 22 min

Figure 2: A coordinate grid illustrating the addition of scaled vectors 3a and 2b using the head-to-tail method to find the resultant 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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