1. Vectors 3: Position Vectors and Scalar Products
Department of Mathematics
University of Leicester

2. Contents Introduction Introduction Position Vector Position Vectors
Scalar Product

3. 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.
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4. Show position vector corresponding to A.
Introduction
Position Vector
Scalar Product
Position Vector A B C O
Show position vector corresponding to A.

5. Show position vector corresponding to B.
Introduction
Position Vector
Scalar Product
Position Vector
Show position vector corresponding to B. A B C O a

6. Show position vector corresponding to C.
Introduction
Position Vector
Scalar Product
Position Vector A B C O a b
Show position vector corresponding to C.

7. Position Vector Next B b C c A a O Introduction Position Vector
Scalar Product
Position Vector A B C O a c b
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8. What is the position vector of this point: ?
Introduction
Position Vector
Scalar Product
What is the position vector of this point: ? x y 3 5

9. Vector between two points
Introduction
Position Vector
Scalar Product
Vector between two points
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. y A
AB= b - a
a = OA
Click here to see this illustrated. B
b = OB x O
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10. y A
a = OA B
b = OB x O

11. y A
a = OA
Click here to repeat. B
Click here to go back.
b = OB x O

12. Distance between two points
Introduction
Position Vector
Scalar Product
Distance between two points x y O A B
b = OB
a = OA
AB= b - a A= B=
The distance between A and B is the magnitude of AB
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13. Questions: Next y A C B x O What is the position vector BC? ( )
Introduction
Position Vector
Scalar Product x y O A B C 6 4 2
Questions:
What is the position vector BC? ( )
What is the distance between A and B?
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14. Scalar Product This is also known as dot product
Introduction
Position Vector
Scalar Product
Scalar Product
This is also known as dot product
Takes two vectors of equal dimension and generates a scalar
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15. Question... What is ? -12 -1 4 Introduction Position Vector
Scalar Product
Question...
-12
What is ? -1 4

16. Scalar Product a b Next y Click here to see a proof of . θ x
Introduction
Position Vector
Scalar Product
Scalar Product y
Click here to
see a proof of .
θ is the angle between the two vectors, at the point where they meet. a b
It’s the smaller angle – not this one: θ x
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17. Look at the triangle the vectors form:
We can use the Cosine Rule to find the angle θ in terms of a and b:
If we take and
we can evaluate...
b-a a b θ

21. Make sure you multiply out the brackets yourself; you should get the following results:
Click here to go back

22. Introduction
Position Vector
Scalar Product
Scalar Product
To calculate the angle we rearrange the Scalar Product formula in the following way
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23. . Next Introduction Position Vector Scalar Product 8 6 4 (Blue) (Pink)2 -8 -6 -4 -2 2 4 6 8 -2
Origin is (210,320) -4 -6 -8
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24. Question... Which of these angles is the angle calculated using ? x y
Introduction
Position Vector
Scalar Product
Question... x y a b θ x y a b θ
Which of these angles is the angle calculated using ? x y a b θ x y a b θ

25. Question... What is in the following diagram: Next y 3 53.13° 58.42° 1
Introduction
Position Vector
Scalar Product
Question...
What is in the following diagram: y
53.13° 3
58.42° 1 θ
78.46 ° x 1 3
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26. Question... What is if the two vectors are at right angles? 1
Introduction
Position Vector
Scalar Product
Question...
What is if the two vectors are at right angles? 1

27. Question... What is ? 6 14 -6 -14 Introduction Position Vector
Scalar Product
Question...
What is ? 6 14 -6
-14

28. An Interesting Dimension
Introduction
Position Vector
Scalar Product
An Interesting Dimension 0D 1D 2D 3D 4D
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29. Introduction
Position Vector
Scalar Product
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
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