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

Contents Introduction Introduction Position Vector Position Vectors Scalar Product

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. Next

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

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

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.

Position Vector Next B b C c A a O Introduction Position Vector Scalar Product Position Vector A B C O a c b Next

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

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 Next

y A a = OA B b = OB x O

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

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 Next

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 -2 2 4 6 Questions: What is the position vector BC? ( ) What is the distance between A and B? Next

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 Next

Question... What is ? -12 -1 4 Introduction Position Vector Scalar Product Question... -12 What is ? -1 4

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 Next

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

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

Introduction Position Vector Scalar Product Scalar Product To calculate the angle we rearrange the Scalar Product formula in the following way Next

. 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 Next

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

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 Next

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

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

An Interesting Dimension Introduction Position Vector Scalar Product An Interesting Dimension 0D 1D 2D 3D 4D Next

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 . Next