Topics
13. Vectors (Enrichment)
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Middle Years

13.02 Addition, subtraction and scalar multiplication

Lesson
Worksheet
Practice
Worksheet
Geometric addition and subtraction of vectors
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}=\begin{bmatrix}4\\\\9\end{bmatrix}, \mathbf{B}=\begin{bmatrix}8\\\\2\end{bmatrix}, \mathbf{C}= \begin{bmatrix}7\\\\5\end{bmatrix}
b
\mathbf{A}=\begin{bmatrix}-6\\\\7\end{bmatrix}, \mathbf{B}=\begin{bmatrix}-4\\\\-8\end{bmatrix}, \mathbf{C}= \begin{bmatrix}2\\\\1\end{bmatrix}.
c
\mathbf{A}=\begin{bmatrix}4.7\\\\-6.2\end{bmatrix}, \mathbf{B}=\begin{bmatrix}7.4\\\\2.3\end{bmatrix}, \mathbf{C}= \begin{bmatrix}-5.1\\\\-3.5\end{bmatrix}.
d
\mathbf{A}=\begin{bmatrix}\dfrac{5}{6}\\\\\dfrac{6}{5}\end{bmatrix}, \mathbf{B}=\begin{bmatrix}-\dfrac{2}{5}\\\\-\dfrac{7}{5}\end{bmatrix}, \mathbf{C}= \begin{bmatrix}\dfrac{3}{2}\\\\-\dfrac{4}{5}\end{bmatrix}.
Scalar multiplication of vectors
8

Let \mathbf{a}=\begin{bmatrix}2\\4\end{bmatrix}. 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}

9

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
10

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}?

11

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
12

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.

13

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.

14

Consider the vectors \mathbf{a} = - 2 \mathbf{i} + 3 \mathbf{j} and \mathbf{b} = 7 \mathbf{i} + 9 \mathbf{j}.

a

Find \mathbf{a} + \mathbf{b}.

b

Find \left|\mathbf{a} + \mathbf{b}\right|.

15

Consider the vectors \mathbf{a} = - \mathbf{i} + 2 \mathbf{j} and \mathbf{b} = - 5 \mathbf{i} - \mathbf{j}.

a

Find \mathbf{a} - \mathbf{b}.

b

Find \left|\mathbf{a} - \mathbf{b}\right|.

16

Consider the vectors \mathbf{a} = 12 \mathbf{i} - 9 \mathbf{j} and \mathbf{b} = - 8 \mathbf{i} + 6 \mathbf{j}.

a

Find \left|\mathbf{a}\right|.

b

Find \left|\mathbf{b}\right|.

c

Find \mathbf{a} + \mathbf{b}.

d

Find \left|\mathbf{a} + \mathbf{b}\right|.

e

Is \left|\mathbf{a}\right| + \left|\mathbf{b}\right| = \left|\mathbf{a} + \mathbf{b}\right|?

17

Let \mathbf{a} = 3 \mathbf{i} + 4 \mathbf{j} and \mathbf{b} = 6 \mathbf{i} + 8 \mathbf{j}.

a

Find \left|\mathbf{a}\right|.

b

Find \left|\mathbf{b}\right|.

c

Find 5 \left|\mathbf{b}\right|.

d

Find \left| 2 \mathbf{a}\right|.

e

Find \left| 4 \mathbf{a} - 3 \mathbf{b}\right|.

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