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Vectors 3:Position Vectors and Scalar ProductsDepartment of MathematicsUniversity of Leicester
Contents
Position Vectors
Introduction Introduction
Position Vector
Scalar Product
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
Position Vector
A
B
C
O
IntroductionPosition Vector
Scalar Product
Show position vector
corresponding to A.
Position Vector
A
B
C
Oa
IntroductionPosition Vector
Scalar Product
Show position vector
corresponding to B.
Position Vector
A
B
C
Oa
b
IntroductionPosition Vector
Scalar Product
Show position vector
corresponding to C.
Position Vector
A
B
C
Oa
c
b
IntroductionPosition Vector
Scalar Product
Next
What is the position vector of this point: ?
IntroductionPosition Vector
Scalar Product
5
3
5,3
yx 53
x
y
3
5
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.
x
y
O
A
B
b = OB
a = OA
x
y
O
A
B
b = OB
a = OAClick here to repeat.
a b
Click here to go back.
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
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
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
Question...
2
3
2
1
IntroductionPosition Vector
Scalar Product
-1
-12
4
What is ?
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:
• 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
cos2222 babaab
......continue
cos2222 babaab
222cos2 abbaba
......continue
cos2222 babaab
222cos2 abbaba
23
22
21
23
22
21cos2 bbbaaaba
233
222
211 )()()( ababab
......continue
332211 222cos2 babababa
332211cos babababa
Make sure you multiply out the brackets yourself; you should get the following results:
Click here to go back
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
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
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
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 °
What is if the two vectors are at right angles?
Question...
ba.
IntroductionPosition Vector
Scalar Product
0
1
ba
Question...
IntroductionPosition Vector
Scalar Product
6
-6
2
3
2
,
2
4
1
ba
What is ?ba.14
-14
An Interesting Dimension
0D
1D
2D
3D
4D
IntroductionPosition Vector
Scalar Product
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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
