Transcript M1.2 Vectors in mechanics

AS-Level Maths:
Mechanics 1
for Edexcel
M1.2 Vectors in
mechanics
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Definition of a vector
Contents
Definition of a vector
Magnitude of a vector
Unit vectors
Multiplying vectors
Addition and subtraction of vectors
Examination-style questions
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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.
B
This vector could be represented
as: a, a or AB.
A
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.
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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.
The vectors i, j and k are unit vectors in the positive
directions of the x, y and z axes respectively.
A vector can also be represented in column form
a 
 ÷
 a
as  ÷÷ or  b ÷÷ .
b
 ÷
c ÷
 
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Position vectors
The position vector of a point A is the vector OA where
O is the origin.
r is often used to denote a position vector.
For example, if OA = 3i + 2j – k, we might say
r = 3i + 2j – k.
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Magnitude of a vector
Contents
Definition of a vector
Magnitude of a vector
Unit vectors
Multiplying vectors
Addition and subtraction of vectors
Examination-style questions
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Magnitude of a vector
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.
If x = ai + bj then x = a 2  b 2
x
b
Similarly, in three dimensions,
if x = ai + bj + ck
a
then x = a 2  b 2  c 2
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Magnitude questions
Find the magnitude of the following vectors:
a) 3i + 2j
b) 4i – 6j + k
c) -i + 3j – 5k
d)
 - 3

÷

÷

÷
 5
e)
 2

÷

÷
 -1÷

÷
 3÷


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Magnitude solutions
a) Magnitude = 32  22  9  4  13
b) Magnitude = 42  ( -6)2  12  16  36  1  53
2
2
2
(
1)

3

(
5)
 1  9  25  35
c) Magnitude =
d) Magnitude = ( -3)2  52  9  25  34
2
2
2
e) Magnitude = 2  ( -1)  3  4  1  9  14
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Unit vectors
Contents
Definition of a vector
Magnitude of a vector
Unit vectors
Multiplying vectors
Addition and subtraction of vectors
Examination-style questions
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Unit vectors
A unit vector is a vector of magnitude 1.
If v is a vector then the corresponding unit vector is
represented by v̂ .
A unit vector in the direction of a given vector is found by
v
dividing the vector by the magnitude: vˆ =
v
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Unit vector questions
Find a unit vector in the direction of the following vectors:
a) 4i – 2j
b) 3i – j + 4k
c) –3i + 5j – 2k
d)
e)
 3
 ÷
 ÷
 ÷
6
 - 4

÷

÷
 0÷

÷
 3÷


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Unit vector solutions
a) Magnitude  20  2 5 unit vector 
1
 4i - 2 j
2 5
1
b) Magnitude  26 unit vector 
3i - j  4k 
26
1
c) Magnitude  38 unit vector 
 -3i  5 j - 2k 
38
d) Magnitude  45  3 5  unit vector 
3

 1 
5
 3 5

6
  2





5

 3 5
 -4 
 5
e) Magnitude  25  5 unit vector   0 


3 
 5 
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Multiplying vectors
Contents
Definition of a vector
Magnitude of a vector
Unit vectors
Multiplying vectors
Addition and subtraction of vectors
Examination-style questions
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Multiplication of a vector by a scalar
To multiply a vector by a scalar, multiply each component of
the vector by the scalar.
3(2i – 5j + k) = 6i – 15j + 3k
 0  0 
-5  -6    30 
 2   -10 
  

If the scalar is greater than 1, the resulting vector is larger
than the original vector and parallel to it.
If the scalar is less than 1, the resulting vector is smaller
than the original vector and parallel to it.
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Parallel vectors
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.
Two vectors are parallel if one is a scalar multiple of the other:
3i – 3j + 4k and –9i + 9j – 12k are parallel vectors since
–9i + 9j – 12k = –3(3i – 3j+ 4k)
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Addition and subtraction of vectors
Contents
Definition of a vector
Magnitude of a vector
Unit vectors
Multiplying vectors
Addition and subtraction of vectors
Examination-style questions
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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 
     
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 35-2 
 -2 - 3  3  


 1 4 


 6
 -2 
 
 5
 
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Triangle law of addition
To add vectors using the triangle law, one vector is placed
on the end of another. The resultant vector is then shown.
Draw the vector representing a + b.
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a
b
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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.
a
Draw the vector representing a + b.
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b
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Examination-style questions
Contents
Definition of a vector
Magnitude of a vector
Unit vectors
Multiplying vectors
Addition and subtraction of vectors
Examination-style questions
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Examination-style question 1
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?
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Solution 1
If the river is 24 m wide and it takes 16 seconds to cross, the
resultant speed is 1.5 ms–1 (24  16).
Draw a triangle of velocities:
2
v
1.5
x°
v2 = 22 + 1.52  v = 2.5
tan x = 2  1.5  x = 53.1°
Therefore the boy swims at a speed of
2.5 ms–1 at an angle of 36.9° to the bank.
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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°.
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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.
a = t(2i + 4j) = 2ti + 4tj
b = 5i + t(–2i + 5j) = (5 – 2t)i + 5tj
c) At what time will A be due north of B?
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.
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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
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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.
Therefore, 25 = 17t2 – 40t + 25
 17t2 – 40t = 0
 t(17t – 40) = 0
 t = 0 or t = 40  17
40  17 = 2.35 (to 3 s.f.), which is equivalent to 2 hours 21
minutes (to the nearest minute).
Therefore the boats are again 5 km apart at 4:21pm.
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