iGCSE (2021 Edition)

# 10.01 Introduction to vectors

Worksheet
Column vectors
1

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

a
b
2

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

a

\overrightarrow{AB}

b

\overrightarrow{BC}

c

\overrightarrow{AC}

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
b
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).

Magnitude of a vector
8

Find the magnitude of vector a:

9

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

10

Find the magnitude of vector b:

11

Consider the plotted path:

a

Calculate the displacement of the path.

b

Calculate the distance of the path.

c

12

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.

13

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

14

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}

15

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

16

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

17

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

18

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.

19

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

20

Consider the vector 12 \mathbf{i} - 16 \mathbf{j}. Find the magnitude of the vector.

21

Consider the vector v = 12 \mathbf{i} - 9 \mathbf{j}. Find the magnitude of the vector.

22

Suppose \mathbf{a} = 9 \mathbf{i} + 12 \mathbf{j} and \mathbf{b} = 6 \mathbf{i} + 8 \mathbf{j}. Find \left|\mathbf{b}\right|.

23

Find the magnitude of the vector 12 \mathbf{i} + 16 \mathbf{j}.

Direction of a vector
24

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.

25

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

26

Let \mathbf{U} = \left( - 2 , - 4 \right) be a vector. Determine the direction angle \theta of the vector correct to the nearest degree, where 0 \degree \leq \theta < 360 \degree.

27

Let \mathbf{U} = \left( - \dfrac{\sqrt{3}}{2} , \dfrac{1}{2}\right) be a vector.

a

Find the magnitude of the vector \mathbf{U}.

b

Determine the direction angle \theta of vector \mathbf{U}, where 0 \degree \leq \theta < 360 \degree.

28

Let \mathbf{U} = 5 \mathbf{i} - 3 \mathbf{j} be a vector.

a

Find the exact magnitude of the vector \mathbf{U}.

b

Determine the direction angle \theta of vector \mathbf{U} in degrees, where 0 \degree \leq \theta < 360 \degree. Round you answer to two decimal places.

29
a

Find the magnitude \left|\mathbf{u} + \mathbf{v}\right| to four decimal places.

b

Find the direction angle \theta, in degrees, of \mathbf{u} + \mathbf{v}. Round your answer to four decimal places.

c

Write the vector \mathbf{u} in the form \left(a, b\right), where a and b are to four decimal places.

d

Write the vector \mathbf{v} in the form \left(a, b\right), where a and b are to four decimal places.

e

Write the vector \mathbf{u} + \mathbf{v} in the form \left(a, b\right), where a and b are to four decimal places.

f

Find the magnitude of the vector \\ \mathbf{u} + \mathbf{v} = \left( - 6.2116 , 11.5911\right). Round your answer to four decimal places.

g

Find the direction angle \theta, in degrees, of the vector \mathbf{u} + \mathbf{v} = \left( - 6.2116 , 11.5911\right). Round your answer to four decimal places.

### Outcomes

#### 0606C13.1

Use vectors in any form, e.g. (a b), AB, p, ai - bj.

#### 0606C13.2

Know and use position vectors and unit vectors.

#### 0606C13.3

Find the magnitude of a vector; add and subtract vectors and multiply vectors by scalars.