Calculating the Magnitude of a Vector

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

1. Identify horizontal and vertical components of vectors on a Cartesian grid as legs of a right-angled triangle
2. Calculate vector magnitudes using Pythagoras theorem on column vectors and coordinate pairs
3. Solve real-world displacement problems by calculating direct distance as vector magnitude

Key vocabulary

Magnitude of a Vector

The overall length or size of a vector quantity regardless of its direction.

The magnitude of vector $\overrightarrow{PQ}$ is calculated as five units long on the grid.

Component of a Vector

The horizontal or vertical displacement value that makes up a column vector.

The horizontal component of vector $\mathbf{v}=\left(\begin{array}{c}5\\ -12\end{array}\right)$ is positive five.

Displacement Vector

A vector representing the shortest straight-line distance and direction from start to finish.

The displacement vector of this path is $\left(\begin{array}{c}9\\ 12\end{array}\right)$ km.

Lesson preview

Slide 1 · Connecting Vector Components to Right-Angled TrianglesOpening · 12 min

The magnitude of a vector represents its length or direct distance. When we plot a vector on a Cartesian grid, its horizontal and vertical components form the two shorter legs of a right-angled triangle. The vector itself forms the hypotenuse. By reading the horizontal change $x$ and vertical change $y$, we apply Pythagoras theorem to calculate the magnitude, written with vertical bars like $\mathrm{\mid }\overrightarrow{PQ}\mathrm{\mid }$. This links coordinate geometry directly to vector length.

• The horizontal change $x$ and vertical change $y$ form perpendicular legs of a right-angled triangle.
• The vector line segment acts as the hypotenuse connecting the starting point to the ending point.
• The magnitude of column vector $\left(\begin{array}{c}x\\ y\end{array}\right)$ is calculated using Pythagoras formula $\sqrt{x^2+y^2}$.
• We write the magnitude of vector $\overrightarrow{PQ}$ using straight vertical lines as $\mathrm{\mid }\overrightarrow{PQ}\mathrm{\mid }$.

Worked example. Calculate horizontal shift: $x=5-1=4$ units. Calculate vertical shift: $y=5-2=3$ units. Write column vector: $\overrightarrow{PQ}=\left(\begin{array}{c}4\\ 3\end{array}\right)$. Apply Pythagoras theorem: $\mathrm{\mid }\overrightarrow{PQ}\mathrm{\mid }=\sqrt{4^2+3^2}=\sqrt{16+9}=\sqrt{25}=5$ units. Answer: The column vector is $\left(\begin{array}{c}4\\ 3\end{array}\right)$ and its magnitude is $5\text{ units}$..

Check question: Points $C(2,3)$ and $D(8,11)$ form vector $\overrightarrow{CD}$. What is the magnitude $\mathrm{\mid }\overrightarrow{CD}\mathrm{\mid }$?

1. 10 units
2. 14 units
3. 100 units
4. $\sqrt{296}\text{ units}$

Show answer

10 units. The horizontal component is $8-2=6$ and the vertical component is $11-3=8$. Magnitude $\mathrm{\mid }\overrightarrow{CD}\mathrm{\mid }=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10\text{ units}$.

Slide 2 · Calculating Vector Magnitudes from Coordinates and Column VectorsModel · 14 min

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• All 3 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