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Vectors 3:Position Vectors and Scalar ProductsDepartment of MathematicsUniversity of Leicester
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Contents
Position Vectors
Introduction Introduction
Position Vector
Scalar Product
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Introduction
• A point can be represented by a position vector that gives its distance and direction from the origin.
• At the point where two vectors meet, we can use an operation called the ‘scalar product’ to find the angle between them.
IntroductionPosition Vector
Scalar Product
Next
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Position Vector
A
B
C
O
IntroductionPosition Vector
Scalar Product
Show position vector
corresponding to A.
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Position Vector
A
B
C
Oa
IntroductionPosition Vector
Scalar Product
Show position vector
corresponding to B.
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Position Vector
A
B
C
Oa
b
IntroductionPosition Vector
Scalar Product
Show position vector
corresponding to C.
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Position Vector
A
B
C
Oa
c
b
IntroductionPosition Vector
Scalar Product
Next
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What is the position vector of this point: ?
IntroductionPosition Vector
Scalar Product
5
3
5,3
yx 53
x
y
3
5
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Vector between two points
x
y
O
A
B
b = OB
a = OAAB= b - a
If we start at the point A, travel along the vector a in the negative direction and then travel along the vector b; this is how we get the vector AB.
IntroductionPosition Vector
Scalar Product
Next
Click here to see this
illustrated.
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x
y
O
A
B
b = OB
a = OA
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x
y
O
A
B
b = OB
a = OAClick here to repeat.
a b
Click here to go back.
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Distance between two points
x
y
O
A
B
b = OB
a = OAAB= b - a
A=
B=
The distance between A and B is the magnitude of AB
2
1
b
b
2
1
a
a
212
211 )()( ababAB
IntroductionPosition Vector
Scalar Product
Next
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Questions:
IntroductionPosition Vector
Scalar Product
x
y
O
A
B
C
6
4
2
-2 2 4 6
What is the position vector BC? ( )What is the distance between A and B?
Next
Check Answers
Show Answers
Clear Answers
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Scalar Product
• This is also known as dot product
• Takes two vectors of equal dimension and generates a scalar
332211
3
2
1
3
2
1
bababa
b
b
b
a
a
a
ba
IntroductionPosition Vector
Scalar Product
Next
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Question...
2
3
2
1
IntroductionPosition Vector
Scalar Product
-1
-12
4
What is ?
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Scalar Productcos332211 bababababa
x
y
a
bθ
IntroductionPosition Vector
Scalar Product
Next
cosbaba
Click here to see a proof of
.θ is the angle
between the two vectors, at the point where they meet.
It’s the smaller angle – not this one:
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• Look at the triangle the vectors form:
a
bθ
b-a
We can use the Cosine Rule to find the angle θ in terms of a and b:
If we take and
we can evaluate...
cos2222 babaab
2
1
a
aa
2
1
b
bb
......continue
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cos2222 babaab
......continue
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cos2222 babaab
222cos2 abbaba
......continue
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cos2222 babaab
222cos2 abbaba
23
22
21
23
22
21cos2 bbbaaaba
233
222
211 )()()( ababab
......continue
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332211 222cos2 babababa
332211cos babababa
Make sure you multiply out the brackets yourself; you should get the following results:
Click here to go back
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Scalar Product
cosbaba
cos332211 bababababa
To calculate the angle we rearrange the Scalar Product formula in the following way
baba1cos
IntroductionPosition Vector
Scalar Product
Next
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2
-2
4
6
8
-6
-8
-4
0-2-6 -4 2 4 6 8-8
(Blue) (Pink)
y
x
IntroductionPosition Vector
Scalar Product
Next
.
Show angle between them
Clear
Draw vectors
Calculate angle
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Question...
IntroductionPosition Vector
Scalar Product
Which of these angles is the angle calculated using ?
cosbaba
x
y
a
bθ
x
y
a
b
θx
y
a
bθ
x
y
a
b
θ
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What is in the following diagram:
Question...
IntroductionPosition Vector
Scalar Product
Next
x
y
θ
3
1
1 3
53.13°
58.42°
78.46 °
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What is if the two vectors are at right angles?
Question...
ba.
IntroductionPosition Vector
Scalar Product
0
1
ba
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Question...
IntroductionPosition Vector
Scalar Product
6
-6
2
3
2
,
2
4
1
ba
What is ?ba.14
-14
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An Interesting Dimension
0D
1D
2D
3D
4D
IntroductionPosition Vector
Scalar Product
Next
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Conclusion
• Position Vectors are used to describe the size and position of a vector.
• Scalar Multiplication is used to find the angle between vectors.
• Two vectors are at right angles if and only if .
IntroductionPosition Vector
Scalar Product
Next
0. ba
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