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Www.le.ac.uk Vectors 3: Position Vectors and Scalar Products Department of Mathematics University of Leicester.

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

2 Contents Position Vectors Introduction Position Vector Scalar Product

3 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 VectorScalar Product Next

4 Position Vector A B C O IntroductionPosition VectorScalar Product Show position vector corresponding to A.

5 Position Vector A B C O a IntroductionPosition VectorScalar Product Show position vector corresponding to B.

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

7 Position Vector A B C O a c b IntroductionPosition VectorScalar Product Next

8 What is the position vector of this point: ? IntroductionPosition VectorScalar Product x y 3 5

9 Vector between two points x y O A B b = OB a = OA AB= 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 VectorScalar Product Next Click here to see this illustrated.

10 x y O A B b = OB a = OA

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

12 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 IntroductionPosition VectorScalar Product Next

13 Questions: IntroductionPosition VectorScalar Product x y O A B C What is the position vector BC? ( ) What is the distance between A and B? Next

14 Scalar Product This is also known as dot product Takes two vectors of equal dimension and generates a scalar IntroductionPosition VectorScalar Product Next

15 Question... IntroductionPosition VectorScalar Product What is ?

16 Scalar Product x y a b θ IntroductionPosition VectorScalar Product Next 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:

17 Look at the triangle the vectors form: a b θ b-ab-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...

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

22 Scalar Product To calculate the angle we rearrange the Scalar Product formula in the following way IntroductionPosition VectorScalar Product Next

23 (Blue) (Pink) IntroductionPosition VectorScalar Product Next.

24 Question... IntroductionPosition VectorScalar Product Which of these angles is the angle calculated using ? x y a b θ x y a b θ x y a b θ x y a b θ

25 What is in the following diagram: Question... IntroductionPosition VectorScalar Product Next x y θ ° ° °

26 What is if the two vectors are at right angles? Question... IntroductionPosition VectorScalar Product 0 1

27 Question... IntroductionPosition VectorScalar Product 6 -6 What is ?

28 An Interesting Dimension 0D 1D 2D 3D 4D IntroductionPosition VectorScalar Product Next

29 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 VectorScalar Product Next

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