Grade 10 · Core Mathematics · Lesson 7 of 21

Adding Vectors Geometrically

A Grade 10 Core Mathematics CBE lesson from the sub-strand Vectors I. In it, learners use equivalent vectors to add two vectors geometrically.

Strand Measurements and GeometrySub-strand Vectors ILesson code 10CM-28-L740 minutes · 2 slides

KICD learning outcome (e)

Add vectors in different situations.

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

Use equivalent vectors to add two vectors geometrically.
Apply the head-to-tail triangle method to find the resultant vector.
Demonstrate that geometric vector addition is commutative.

Key vocabulary

Equivalent Vectors: Vectors that have the exact same magnitude and point in the same direction.; We draw an equivalent vector to shift a displacement vector to a new start point.
Resultant Vector: The single vector that represents the combined effect of two or more vectors.; The vector $a + b$ is the resultant vector of $a$ and $b$ .
Triangle Method: A geometric technique of placing vectors head-to-tail to form a triangle sum.; The triangle method allows geometric addition of vectors without using coordinate components.

Lesson preview

Slide 1 · Head-to-Tail Vector AdditionModel · 18 min

Figure 1: A vector diagram demonstrating how to geometrically add vector a and vector b by placing them head-to-tail to find the resultant.

When adding vectors in different situations geometrically, we place the vectors head-to-tail. To add vector $b$ to vector $a$ , we trace vector $a$ and then draw an equivalent vector to $b$ starting at the arrowhead of $a$ . Completing the triangle by drawing a directed line segment from the initial tail to the final head yields the resultant sum $a + b$ . Use Figure 1 to visualise how connecting vectors head-to-tail forms a triangle representing the resultant sum.

Adding vectors in different situations requires placing vectors in a head-to-tail sequence on paper.
To begin, trace vector $a$ accurately maintaining its length and original direction.
Draw an equivalent vector to $b$ starting at the head of vector $a$ .
Complete the triangle to represent the resultant sum vector from start to end.

Worked example. Draw vector

a

as a horizontal arrow of length

4 km

pointing East. From the head of vector

a

, draw an equivalent vector to

b

pointing North with a length of

3 km

. Complete the triangle by drawing the resultant vector

a + b

from the tail of

a

to the head of

b

. Calculate magnitude using Pythagoras' theorem:

∣ a + b ∣ = \sqrt{4^{2} + 3^{2}} = \sqrt{16 + 9} = \sqrt{25} = 5 km

. Calculate angle

θ

North of East:

\tan θ = \frac{3}{4} = 0.75

, so

θ \approx {36.87}^{\circ}

, giving a bearing of

090^{\circ} - {36.87}^{\circ} = {053.13}^{\circ}

. Answer: The resultant vector $a + b$ has a magnitude of $5 km$ on a bearing of ${053.13}^{\circ}$ ..

Check question: A boat travels along vector $p$ of $6 km$ East and then along vector $q$ of $8 km$ South. What is the magnitude of the resultant vector $p + q$ ?

$14 km$
$10 km$
$2 km$
$100 km$

Show answer

$10 km$ . Constructing the head-to-tail triangle gives a right-angled triangle with legs of $6 km$ and $8 km$ . Using Pythagoras' theorem, the resultant magnitude is $∣ p + q ∣ = \sqrt{6^{2} + 8^{2}} = \sqrt{36 + 64} = \sqrt{100} = 10 km$ .

Slide 2 · Adding Vectors in Reverse OrderPractice · 22 min

Figure 2: Two vector addition diagrams comparing the resultant vector of u added to v with the resultant of v added to u.

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Homework and the other 20 lessons in this sub-strand

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Adding Vectors Geometrically

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

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 21 lessons in Vectors I