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AS-Level Maths: Mechanics 1for Edexcel

M1.2 Vectors in mechanics

For more detailed instructions, see the Getting Started presentation.


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Definition of a vector


Magnitude of a vector

Unit vectors

Multiplying vectors

Addition and subtraction of vectors

Examination-style questions


Definition

Vectors are quantities that are completely described by a scalar magnitude and a direction. Displacement, velocity, acceleration and force can all be vectors – they can be described fully by their magnitude and direction.

Vectors are often represented by a directed line segment.

A

B

A bold letter is used to represent vectors in text books and on exam papers. When writing by hand, the convention is to underline the letter.

This vector could be represented

as: a, a or AB.


Column and component form

A vector can be represented in component form as ai + bj in two dimensions, or ai + bj + ck in three dimensions.

A vector can also be represented in column form

as or .

ba

c

b

a

The vectors i, j and k are unit vectors in the positive directions of the x, y and z axes respectively.


Position vectors

r is often used to denote a position vector.

For example, if OA = 3i + 2j – k, we might say

r = 3i + 2j – k.

The position vector of a point A is the vector OA where O is the origin.


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Unit vectors

Multiplying vectors





The magnitude of a vector is the size of the vector. When representing a vector by a directed line segment, the magnitude of the vector is represented by its length.

The magnitude of a vector is equal to . a

The magnitude of a vector is found using Pythagoras’ Theorem.

a

bx

If x = ai + bj then = x

Similarly, in three dimensions,if x = ai + bj + ck then = x 2 2 2a b c

2 2a b


Magnitude questions

Find the magnitude of the following vectors:

a) 3i + 2j

b) 4i – 6j + k

c) -i + 3j – 5k

5

3d)

3

1

2

e)


Magnitude solutions

2 2 2( 1) 3 ( 5) 1 9 25 35

b) Magnitude =

c) Magnitude =

d) Magnitude =

e) Magnitude =

a) Magnitude = 2 23 2 9 4 13

2 2 24 ( 6) 1 16 36 1 53

2 2( 3) 5 9 25 34

2 2 22 ( 1) 3 4 1 9 14


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Unit vectors



Unit vectors

Multiplying vectors




Unit vectors

A unit vector is a vector of magnitude 1.

A unit vector in the direction of a given vector is found by dividing the vector by the magnitude: v

vv =ˆ

If v is a vector then the corresponding unit vector is represented by .v̂


Unit vector questions

Find a unit vector in the direction of the following vectors:

3

0

4

6

3

a) 4i – 2j

b) 3i – j + 4k

c) –3i + 5j – 2k

d)

e)


Unit vector solutions

1b) Magnitude 26 unit vector 3 4

26 i j k

1c) Magnitude 38 unit vector 3 5 2

38 i j k

d) Magnitude 45 3 5 unit vector

3 13 5 5

6 2

53 5

45

e) Magnitude 25 5 unit vector 0

35

1a) Magnitude 20 2 5 unit vector 4 2

2 5 i j


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Multiplying vectors



Unit vectors

Multiplying vectors




Multiplication of a vector by a scalar

If the scalar is greater than 1, the resulting vector is larger than the original vector and parallel to it.

3(2i – 5j + k) = 6i – 15j + 3k

0 0

5 6 30

2 10

To multiply a vector by a scalar, multiply each component of the vector by the scalar.

If the scalar is less than 1, the resulting vector is smaller than the original vector and parallel to it.


Parallel vectors

Two vectors are parallel if one is a scalar multiple of the other:

For vectors to be equal, the i, j and k components must be equal.

One vector is the negative of another if each of their corresponding components have opposite signs.

3i – 3j + 4k and –9i + 9j – 12k are parallel vectors since

–9i + 9j – 12k = –3(3i – 3j+ 4k)


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Unit vectors

Multiplying vectors




Addition and subtraction

Adding and subtracting vectors in component form is simply a case of adding and subtracting the i, j and k components separately.

Add the following vectors: 3i + 5j – k, 2i + 3k and –4i + 3j + k.

(3i + 5j – 2k) + (2i + 3k) + (–4i + 3j + k) = i + 8j + 2k

3 5 2

Add the following vectors : 2 , 3 and 3

1 0 4

3 5 2

2 3 3

1 0 4

3 5 2

2 3 3

1 4

6

2

5


Triangle law of addition

abDraw the vector representing a + b.

To add vectors using the triangle law, one vector is placed on the end of another. The resultant vector is then shown.


Parallelogram law of addition

To add two vectors, a and b, using the parallelogram law, the vectors are first placed so that they both start from the same fixed point. a is then put on the end of b and b is placed at the end of a. A parallelogram is now formed – the diagonal of this is the vector a + b.

Draw the vector representing a + b.a

b


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Unit vectors

Multiplying vectors




A boy swims across a river from a point A on one bank to a point B on the other. A and B are directly opposite each other and the river is 24 metres wide. The river is flowing at 2 ms–1 parallel to the banks. If the boy reaches point B after swimming for 16 seconds, at what speed and in what direction did he swim?

Examination-style question 1


Solution 1

Draw a triangle of velocities:

If the river is 24 m wide and it takes 16 seconds to cross, the resultant speed is 1.5 ms–1 (24 16).

v2 = 22 + 1.52

Therefore the boy swims at a speed of 2.5 ms–1 at an angle of 36.9° to the bank.

v

2

1.5x°

v = 2.5 x = 53.1°tan x = 2 1.5


Examination-style question 2a

Two boats A and B are travelling with constant velocities. A travels with a velocity of (2i + 4j) kmh–1 and B travels with a velocity of (–2i + 5j) kmh–1.

a) Find the bearings on which A and B are travelling.

A is moving in the first quadrant at an angle of tan–1 2 to the horizontal. This is an angle of 63.4° (to 3 s.f.), so A is travelling on a bearing of 027°.

B is moving in the second quadrant at an angle of tan–1 2.5 to the horizontal. This is an angle of 68.2° (to 3 s.f.), so B is travelling on a bearing of 338°.


Examination-style question 2b and c

At 2pm, A is at point O and B is 5 km due east of O. At time t hours after 2pm, the position vectors of A and B relative to O are a and b respectively.

b) Find a and b in terms of t, i and j.

c) At what time will A be due north of B?

a = t(2i + 4j) = 2ti + 4tj

When A is due north of B, the i components are equal. Equating i components gives 2t = 5 – 2t 4t = 5 t = 1.25.

Therefore A is due north of B at 3:15pm.

b = 5i + t(–2i + 5j) = (5 – 2t)i + 5tj


Examination-style question 2d

At time t hours after 2pm the distance between the two boats is d km.

d) Show that d2 = 17t2 – 40t + 25

The distance between the two boats is AB. d2 is therefore equal to AB2.

AB = b – a = (5 – 2t)i – 2ti + (5t – 4t)j

= (5 – 4t)i + tj

AB2 = (5 – 4t)2 + t2

d2 = 25 – 40t + 16t2 + t2 = 17t2 – 40t + 25


Examination-style question 2e

e) At 2pm the boats were 5 km apart. Find, correct to the nearest minute, the time when the boats are again 5 km apart.

When the boats are 5 km apart, d2 = 25.

40 17 = 2.35 (to 3 s.f.), which is equivalent to 2 hours 21 minutes (to the nearest minute).

Therefore, 25 = 17t2 – 40t + 25 17t2 – 40t = 0

t(17t – 40) = 0

t = 0 or t = 40 17

Therefore the boats are again 5 km apart at 4:21pm.

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