Vector x moves 4 units right and 7 units down. Vector y moves 7 units right and 4 units down. Vector z moves 4 units right and 7 units down. Which choice correctly identifies the equivalent vectors?

Vectors x and z. Both x and z have the exact same horizontal displacement (4 units right) and vertical displacement (7 units down). This means they have an equal magnitude of √(4^2 + (-7)^2) = √(65) units and point in an identical direction. Vector y has different displacements, meaning it points in a different direction.

Grade 10 · Core Mathematics · Lesson 6 of 21

Equivalent Vectors

A Grade 10 Core Mathematics CBE lesson from the sub-strand Vectors I. In it, learners identify pairs of equivalent vectors on a grid by comparing magnitude and direction.

Strand Measurements and GeometrySub-strand Vectors ILesson code 10CM-28-L640 minutes · 2 slides

KICD learning outcome (d)

Identify equivalent vectors in different situations.

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

Identify pairs of equivalent vectors on a grid by comparing magnitude and direction.
Distinguish between vectors with equal magnitude, vectors in the same direction, and opposite vectors.

Key vocabulary

Equivalent vectors: Vectors that have both equal magnitude and the exact same direction.; Two vectors are equivalent vectors if they share identical length and exact direction.
Magnitude: The length or size of a vector quantity independent of its direction.; The magnitude of a vector moving 3 units right and 4 units up is 5 units.
Opposite vectors: Vectors having equal magnitude but pointing in completely opposite directions.; Vectors with equal length pointing in reverse directions are opposite vectors.

Lesson preview

Slide 1 · Identifying Equivalent Vectors on a GridModel · 18 min

Figure 1: This graph shows three vectors on a coordinate grid, highlighting two equivalent vectors in green to demonstrate they share the exact same magnitude and direction.

Two vectors are equivalent if and only if they have both the same magnitude and the same direction. When examining vectors drawn on a grid, we compare their horizontal and vertical displacements. Two displacement vectors moving along identical horizontal and vertical steps in the same direction are equivalent, regardless of their starting locations on the plane. Use Figure 1 to see how equivalent vectors have identical horizontal and vertical displacements.

Equivalent vectors must possess exactly equal lengths and point in the same direction on a grid.
Vectors with identical horizontal and vertical grid steps in the same direction are equal in magnitude and direction.
Position does not affect vector equivalence; vectors displaced to different starting points can still be equivalent.
We test equivalence by calculating magnitude using Pythagoras' theorem and checking direction using horizontal and vertical displacements.

Worked example. Calculate components for vector

\vec{a}

: 3 units East (right) and 4 units North (up), so magnitude is

\sqrt{3^{2} + 4^{2}} = 5

units, pointing generally North-East slope

\frac{4}{3}

. Calculate components for vector

\vec{b}

4 - 1 = 3

units East and

5 - 1 = 4

units North, giving magnitude

\sqrt{3^{2} + 4^{2}} = 5

units and slope

\frac{4}{3}

. Calculate components for vector

\vec{c}

3 - 0 = 3

units East and

1 - 5 = - 4

units (4 units South), giving magnitude

\sqrt{3^{2} + (- 4)^{2}} = 5

units, but slope

- \frac{4}{3}

(pointing South-East). Compare magnitude and direction:

\vec{a}

and

\vec{b}

have the same magnitude (5 units) and same direction (3 East, 4 North). Answer: Vectors $\vec{a}$ and $\vec{b}$ are equivalent vectors; vector $\vec{c}$ is not equivalent because its direction is different..

Check question: Vector $\vec{x}$ moves 4 units right and 7 units down. Vector $\vec{y}$ moves 7 units right and 4 units down. Vector $\vec{z}$ moves 4 units right and 7 units down. Which choice correctly identifies the equivalent vectors?

Vectors $\vec{x}$ and $\vec{y}$
Vectors $\vec{x}$ and $\vec{z}$
Vectors $\vec{y}$ and $\vec{z}$
All three vectors are equivalent

Show answer

Vectors $\vec{x}$ and $\vec{z}$ . Both $\vec{x}$ and $\vec{z}$ have the exact same horizontal displacement (4 units right) and vertical displacement (7 units down). This means they have an equal magnitude of $\sqrt{4^{2} + (- 7)^{2}} = \sqrt{65}$ units and point in an identical direction. Vector $\vec{y}$ has different displacements, meaning it points in a different direction.

Slide 2 · Comparing Vectors by Magnitude and DirectionPractice · 22 min

Figure 2: The image displays four distinct vectors on a grid, each labelled with its length and direction to allow comparisons of their magnitudes and orientations.

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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