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iGCSE (2021 Edition)

25.01 Introduction to vectors

Worksheet
Column vectors
1

Write the vector represented on the following planes as a column vector:

a
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
b
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
2

Consider the following figure and write the column vector for each of the following:

a

\overrightarrow{AB}

b

\overrightarrow{BC}

c

\overrightarrow{AC}

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

For each of the following graphs:

i

Write the vector c as a column vector.

ii

Write the vector d as a column vector.

a
1
2
3
4
5
6
7
8
9
x
1
2
3
4
5
6
7
8
9
y
b
1
2
3
4
5
6
7
8
9
x
1
2
3
4
5
6
7
8
9
y
4

If A = \left( - 2 , 3\right), B = \left(1, 7\right), C = \left( - 9 , 6\right) and O is the origin, express each of the following as a column vector:

a

\overrightarrow{OB}

b

\overrightarrow{AC}

c

\overrightarrow{AO}

d

\overrightarrow{CB}

5

Plot the vector \begin{bmatrix}5\\9\end{bmatrix} on a number plane. Use the origin as the starting point of the vector.

6

Let A = \left( - 6 , 2\right), B = \left(1, 4\right), C = \left( - 3 , 5\right) and O be the origin. Plot the following vectors on a number plane:

a

\overrightarrow{OC}

b

\overrightarrow{AB}

c

\overrightarrow{BO}

d

\overrightarrow{OC}

7

For each of the following sets of points:

i

Plot the vectors \overrightarrow{AB} and \overrightarrow{CD} on on a number plane.

ii

State whether the two vectors are equivalent.

a

A\left(5, 4\right), B\left(5, 9\right), C\left( - 4 , - 2 \right) and D\left( - 4 , 3\right).

b

A\left( - 4 , 2\right), B\left( - 6 , 6\right), C\left( - 4 , - 2 \right) and D\left( - 6 , 2\right).

c

A\left(3, - 4 \right), B\left(3, - 6 \right), C\left(5, 3\right) and D\left(5, 5\right).

8

State whether each of the following is a vector quantity:

a

A force of 8 \text{ N} acting horizontally right.

b

A displacement of 4 \text{ m} along the line joining A and B.

c

A mass of 1 \text{ kg}

d

A time of 1 second.

9

What is the opposite of the vector representing 800 km south?

Magnitude of a vector (Extended)
10

Find the magnitude of vector a:

1
2
3
4
5
6
7
8
9
x
1
2
3
4
5
6
7
8
9
y
11

Find the magnitude of the vector between points A and B:

1
2
3
4
5
6
7
8
9
x
1
2
3
4
5
6
7
8
9
y
12

Find the magnitude of vector b:

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

Consider the plotted path:

a

Calculate the displacement of the path.

b

Calculate the distance of the path.

c

Is the displacement equal to the distance? Explain your answer.

1
2
3
4
5
6
7
8
9
10
11
12
x
1
2
3
4
5
6
7
8
9
10
11
12
y
14

Consider the vector defined by the directed line segment from \left( - 5 , 3\right) to \left(1, - 5 \right).

a

Plot the vector.

b

Find the magnitude of the vector.

15

Two vectors that are parallel can have different magnitudes. Is it true or false?

16

Find the exact magnitudes of the following vectors

a
\begin{bmatrix}-7\\\,\,\,0\end{bmatrix}
b
\begin{bmatrix}5\\-7\end{bmatrix}
c

\begin{bmatrix}8\\15\end{bmatrix}

d

\begin{bmatrix}-8\\-15\end{bmatrix}

e

\begin{bmatrix}6\\0\end{bmatrix}

f

\begin{bmatrix}-6\\0\end{bmatrix}

17

Find the exact magnitude of the vector \left(4, 8\right).

18

Find the magnitude of the vector \left(12, - 4 , 3\right).

19

If a = \left( 3 x, 4 x\right) and \left|a\right| = 15, find the value of x.

20

A pilot flies 12 km north and then 16 km east. Find the magnitude of the vector created from her initial position to the final position.

21

A boat travels 12 km west and then 5 km south. Determine the distance of the boat from its initial position.

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Outcomes

0580C7.1A

Describe a translation by using a vector represented by e.g. exyo, AB or a.

0580E7.1A

Describe a translation by using a vector represented by e.g. exyo, AB or a.

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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