Representing Vectors Geometrically

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

1. Draw directed line segments horizontally, vertically, and in compass directions to represent vector quantities geometrically.
2. Identify the magnitude and direction of a vector from its geometric representation.

Key vocabulary

Directed line segment

A straight line segment with an arrow indicating a specific direction from start to end.

The horizontal arrow drawn from point A to point B forms a directed line segment.

Terminal point

The ending point of a directed line segment where the arrow head is placed.

Point B is the terminal point of the vector starting at initial point A.

Magnitude

The numerical size or length of a vector quantity represented by line length.

A vector represented by a line segment of length $4.2\text{ cm}$ has magnitude $4.2\text{ cm}$.

Lesson preview

Slide 1 · Geometric Representation of Horizontal and Vertical VectorsOpening · 18 min

A vector quantity has both magnitude and direction, so we represent it geometrically using a directed line segment. The initial point shows where the vector starts, while the terminal point shows where it ends. The straight distance between these two points measured with a ruler gives the magnitude, while an arrowhead along the line explicitly indicates the direction.

• A directed line segment uses a straight line with an arrowhead to show vector direction.
• The initial point marks the start of the vector and the terminal point marks its end.
• A horizontal line segment drawn left to right represents horizontal vector displacement.
• A vertical line segment drawn from bottom to top shows vertical vector movement.

Worked example. 1. For vector $MN$, mark initial point $M$. Draw a horizontal line of length $5.2\text{ cm}$ to terminal point $N$, placing an arrowhead at $N$ pointing East. 2. For vector $GH$, mark initial point $G$. Draw a vertical line extending downwards by $3.8\text{ cm}$ to terminal point $H$, with an arrowhead at $H$ pointing South. 3. State the magnitude and direction: $MN$ has length $5.2\text{ cm}$ East; $GH$ has length $3.8\text{ cm}$ South. Answer: Vector $MN$ is a horizontal arrow of length $5.2\text{ cm}$ pointing East; vector $GH$ is a vertical arrow of length $3.8\text{ cm}$ pointing South..

Check question: A student draws a vertical line from point $P$ up to point $Q$ of length $4.5\text{ cm}$ with an arrow at $Q$. What are the magnitude, initial point, and direction of this vector?

1. Magnitude $4.5\text{ cm}$, initial point $P$, direction North
2. Magnitude $4.5\text{ cm}$, initial point $Q$, direction North
3. Magnitude $4.5\text{ cm}$, initial point $P$, direction South
4. Magnitude $4.5\text{ cm}$, initial point $Q$, direction South

Show answer

Magnitude $4.5\text{ cm}$, initial point $P$, direction North. The length of $4.5\text{ cm}$ gives the magnitude. Since the line is drawn from $P$, point $P$ is the starting or initial point. The arrow is at $Q$ pointing straight up, which represents the North direction.

Slide 2 · Representing Vectors in Compass DirectionsPractice · 22 min

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