If u = pmatrix 3 \ 4 pmatrix and v = pmatrix 9 \ 1 pmatrix, calculate the magnitude |u + v|.

13 units. First, find the resultant vector by adding the corresponding components: u + v = pmatrix 3+9 \ 4+1 pmatrix = pmatrix 12 \ 5 pmatrix. Then, calculate its magnitude using Pythagoras' theorem: |u + v| = √(12^2 + 5^2) = √(144 + 25) = √(169) = 13 units.

Grade 10 · Core Mathematics · Lesson 16 of 21

Working Out Vector Magnitudes in Different Situations

A Grade 10 Core Mathematics CBE lesson from the sub-strand Vectors I. In it, learners calculate the magnitude of combined vectors and linear scalar multiples.

Strand Measurements and GeometrySub-strand Vectors ILesson code 10CM-28-L1640 minutes · 3 slides

KICD learning outcome (i)

Work out magnitude of a vector in different situations.

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

Calculate the magnitude of combined vectors and linear scalar multiples.
Determine column displacement vectors from coordinate points and calculate their magnitudes.
Apply vector magnitudes to solve real-life direct distance and displacement problems.

Key vocabulary

Resultant vector: The single vector obtained by combining two or more vectors through addition or subtraction.; Vector $(\begin{array}{c} 3 \\ 4 \end{array})$ is the resultant vector of $(\begin{array}{c} 1 \\ 1 \end{array})$ plus $(\begin{array}{c} 2 \\ 3 \end{array})$ .
Displacement vector: A vector showing the straight-line distance and direction from an initial to a final position.; The displacement vector from $A (1, 2)$ to $B (4, 6)$ is $(\begin{array}{c} 3 \\ 4 \end{array})$ .
Position vector: A vector that represents the position of a point relative to the origin.; Point $P (4, 3)$ has position vector $\vec{O P} = (\begin{array}{c} 4 \\ 3 \end{array})$ relative to the origin.

Lesson preview

Slide 1 · Magnitudes of Combined Column VectorsOpening · 12 min

Figure 1: A graph showing the addition of two column vectors, a and b, to form the resultant vector c.

When two or more column vectors are combined through addition, subtraction, or scalar multiplication, we first determine the resultant column vector by performing component-wise operations. Once the single resultant vector is written in the form $(\begin{array}{c} x \\ y \end{array})$ , its magnitude is calculated using Pythagoras' theorem $∣ v ∣ = \sqrt{x^{2} + y^{2}}$ . This gives the direct straight-line distance or overall length of the combined vector. Use Figure 1 to see how the magnitude of a combined vector is calculated from its components.

Combine column vectors by operating on horizontal and vertical components separately before calculating the overall magnitude.
Scalar multiplication multiplies both top and bottom components of the column vector before vector addition.
The magnitude of resultant vector $(\begin{array}{c} x \\ y \end{array})$ is calculated using $∣ v ∣ = \sqrt{x^{2} + y^{2}}$ .
Magnitudes are positive scalar values representing the true geometric length of the combined vector.

Worked example. 1. Add components to find

a + b = (\begin{array}{c} 6 + 2 \\ 2 + (- 8) \end{array}) = (\begin{array}{c} 8 \\ - 6 \end{array})

. 2. Calculate magnitude

∣ a + b ∣ = \sqrt{8^{2} + (- 6)^{2}} = \sqrt{64 + 36} = \sqrt{100} = 10 units

. 3. Evaluate linear combination

2 a + b = 2 (\begin{array}{c} 6 \\ 2 \end{array}) + (\begin{array}{c} 2 \\ - 8 \end{array}) = (\begin{array}{c} 12 + 2 \\ 4 - 8 \end{array}) = (\begin{array}{c} 14 \\ - 4 \end{array})

. 4. Calculate magnitude

∣ 2 a + b ∣ = \sqrt{14^{2} + (- 4)^{2}} = \sqrt{196 + 16} = \sqrt{212} \approx 14.56 units

. Answer: (a) $a + b = (\begin{array}{c} 8 \\ - 6 \end{array})$ , $∣ a + b ∣ = 10 units$ ; (b) $2 a + b = (\begin{array}{c} 14 \\ - 4 \end{array})$ , $∣ 2 a + b ∣ \approx 14.56 units$ ..

Check question: If $u = (\begin{array}{c} 3 \\ 4 \end{array})$ and $v = (\begin{array}{c} 9 \\ 1 \end{array})$ , calculate the magnitude $∣ u + v ∣$ .

$13 units$
$17 units$
$\sqrt{17} units$
$169 units$

Show answer

$13 units$ . First, find the resultant vector by adding the corresponding components: $u + v = (\begin{array}{c} 3 + 9 \\ 4 + 1 \end{array}) = (\begin{array}{c} 12 \\ 5 \end{array})$ . Then, calculate its magnitude using Pythagoras' theorem: $∣ u + v ∣ = \sqrt{12^{2} + 5^{2}} = \sqrt{144 + 25} = \sqrt{169} = 13 units$ .

Slide 2 · Vector Magnitudes in Real-Life ContextsModel · 14 min

Figure 2: A diagram illustrating real-world movements converted into horizontal and vertical displacement vectors to find the overall distance.

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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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