Topics
25. Vectors
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iGCSE (2021 Edition)

25.02 Addition, subtraction and scalar multiplication

Lesson
Worksheet
Practice
Worksheet
Geometric addition and subtraction of vectors (Extended)
1

Consider the following graph and identify which vector is the result of \mathbf{a} + \mathbf{b}:

x
y
2

Consider the graph of vectors \mathbf{a} and \mathbf{b}. Plot the result of \mathbf{a} + \mathbf{b}.

x
y
3

Consider the following graphs of \mathbf{a} and \mathbf{b}:

a

Plot the result of \mathbf{a} + \mathbf{b} in the first graph:

x
y
b

Plot the result of \mathbf{b} + \mathbf{a} in the second graph:

x
y
c

Are the two resulting vectors \mathbf{a} + \mathbf{b} and \mathbf{b} + \mathbf{a} equal?

4

The vectors \mathbf{a}, \mathbf{b} and \mathbf{c} have been graphed as shown:

Plot the result of \mathbf{a} + \mathbf{b}+ \mathbf{c} on the same axes.

x
y
5

Vectors \mathbf{a}, \mathbf{b}, \mathbf{c} and \mathbf{d} have been graphed as shown:

Plot the result of \mathbf{b}+ \mathbf{c} on the same axes.

x
y
Algebraic addition and subtraction of vectors
6

Rewrite \left(6, 5\right) + \left(7, - 4 \right) as a single vector.

7

For each of the following set of vectors, find:

i
\mathbf{A}+\mathbf{B}
ii
\mathbf{B}+\mathbf{C}
iii
\mathbf{A}+\mathbf{B}+\mathbf{C}
a
\mathbf{A}=\left(\begin{matrix}4\\\\9\end{matrix}\right), \mathbf{B}=\left(\begin{matrix}8\\\\2\end{matrix}\right), \mathbf{C}= \left(\begin{matrix}7\\\\5\end{matrix}\right)
b
\mathbf{A}=\left(\begin{matrix}-6\\\\7\end{matrix}\right), \mathbf{B}=\left(\begin{matrix}-4\\\\-8\end{matrix}\right), \mathbf{C}= \left(\begin{matrix}2\\\\1\end{matrix}\right).
c
\mathbf{A}=\left(\begin{matrix}4.7\\\\-6.2\end{matrix}\right), \mathbf{B}=\left(\begin{matrix}7.4\\\\2.3\end{matrix}\right), \mathbf{C}= \left(\begin{matrix}-5.1\\\\-3.5\end{matrix}\right).
d
\mathbf{A}=\left(\begin{matrix}\dfrac{5}{6}\\\\\dfrac{6}{5}\end{matrix}\right), \mathbf{B}=\left(\begin{matrix}-\dfrac{2}{5}\\\\-\dfrac{7}{5}\end{matrix}\right), \mathbf{C}= \left(\begin{matrix}\dfrac{3}{2}\\\\-\dfrac{4}{5}\end{matrix}\right).
Scalar multiplication of vectors
8

Let \mathbf{a}=\left(\begin{matrix}3\\1\end{matrix}\right), find:

a
2\mathbf{a}
b
5\mathbf{a}
c
-1\mathbf{a}
d
-6\mathbf{a}
9

Let \mathbf{a}=\left(\begin{matrix}10\\16\end{matrix}\right), find:

a
3\mathbf{a}
b
-10\mathbf{a}
c
\dfrac{1}{2}\mathbf{a}
d
-\dfrac{1}{4}\mathbf{a}
10

Let \mathbf{a}=\left(\begin{matrix}-4\\9\end{matrix}\right), find:

a
3\mathbf{a}
b
-2\mathbf{a}
c
\dfrac{1}{2}\mathbf{a}
d
-100\mathbf{a}
Scalar multiplication of vectors (Extended)
11

Let \mathbf{a}=\left(\begin{matrix}2\\4\end{matrix}\right). Using the origin as the starting point, plot the following vectors:

a

3 \mathbf{a}

b

- 2 \mathbf{a}

c

\dfrac{1}{2} \mathbf{a}

12

Consider vectors \mathbf{a} and \mathbf{b} plotted on the graph below. Find the column vector for each of the following:

a

\mathbf{a} + \mathbf{b}

b

3 \mathbf{b}

c

4 \mathbf{a}

d

5 \mathbf{b} - \mathbf{a}

e

3 \mathbf{a} - 4 \mathbf{b}

1
2
3
4
5
6
7
8
9
10
x
1
2
3
4
5
6
7
8
9
10
y
13

Consider the vectors shown below:

-12
-8
-4
4
8
12
x
-12
-8
-4
4
8
12
y
-12
-8
-4
4
8
12
x
-12
-8
-4
4
8
12
y
-12
-8
-4
4
8
12
x
-12
-8
-4
4
8
12
y
-12
-8
-4
4
8
12
x
-12
-8
-4
4
8
12
y
a

Which vector is equivalent to 2 \mathbf{a}?

b

Which vector is equivalent to \dfrac{1}{4} \mathbf{b}?

c

Which vector is equivalent to - \mathbf{a}?

d

Which vector is equivalent to - \dfrac{1}{2} \mathbf{d}?

14

Consider the vectors plotted on the graph. Find the column vector for each of the following:

a

3 \mathbf{a}

b

4 \mathbf{c}

c

2 \mathbf{b} + \mathbf{d}

d

\mathbf{c} - \mathbf{d}

e

3 \mathbf{a} - \mathbf{b}

f

- 4 \mathbf{b} + 2 \mathbf{c}

1
2
3
4
5
6
7
8
9
x
1
2
3
4
5
6
7
8
9
y
15

If \mathbf{a} = \left( 2 x, 8 x\right), \mathbf{b} = \left( 3 x, 4 x\right), and \left|\mathbf{a} + \mathbf{b}\right| = 13, find the value of x.

16

If \mathbf{a} = \left( 2 x, 4\right), \mathbf{b} = \left( 3 x, 8 x\right), and \left|\mathbf{a} + \mathbf{b}\right| = 13 x, find the value of x.

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Outcomes

0580C7.1B

Add and subtract vectors. Multiply a vector by a scalar.

0580E7.1B

Add and subtract vectors. Multiply a vector by a scalar.

0580E7.3

Calculate the magnitude of a vector (x y) as x^2 + y^2 . Represent vectors by directed line segments. Use the sum and difference of two vectors to express given vectors in terms of two coplanar vectors. Use position vectors.

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