Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.

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Presentation transcript:

Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester

Contents What is a Vector? Dimensions Basis Drawing Vectors

What is a vector? A vector is a quantity which has direction as well as length They are used to determine the position of one point in space relative to another What is a Vector? DimensionsBasis Drawing Vectors Next

Spot the Difference So, what is the difference between a vector and a scalar? Next What is a Vector? DimensionsBasis Drawing Vectors

Spot the Difference So, what is the difference between a vector and a scalar? A scalar only has magnitude and is independent of direction My car goes at 40mph. Scalar man Next What is a Vector? DimensionsBasis Drawing Vectors

Spot the Difference So, what is the difference between a vector and a scalar? A scalar only has magnitude and is independent of direction A vector has both magnitude AND direction Im going 20 miles West. Vector man My car goes at 40mph. Scalar man Next What is a Vector? DimensionsBasis Drawing Vectors

Spot the Difference Do we use a scalar or vector quantity to represent the top speed of a car? Next What is a Vector? DimensionsBasis Drawing Vectors

Spot the Difference Do we use a scalar or vector quantity to represent the top speed of a car? A: We use a SCALAR ie. the top speed of my car is 120mph. However, if I were to talk about the velocity of the car, this concerns both the speed AND direction in which I am travelling, so a vector is used for velocity. Next What is a Vector? DimensionsBasis Drawing Vectors

Question… Choose whether the following would be measured using vectors or scalars: Temperature Position of the Earth in its orbit around the sun. Distance travelled on a circular walk. Next What is a Vector? DimensionsBasis Drawing Vectors

Dimensions The formal definition of dimension is the minimum number of coordinates used to specify a point in that dimension. A LINE is one dimensional; we only need a single number to get to any point along it (think about the number line). A PLANE or SURFACE is two dimensional; just think about a graph. We need the value along the x and y axis to find a point. Next What is a Vector? DimensionsBasis Drawing Vectors

Dimensions We live in a three dimensional world; can you give a reason why? Next What is a Vector? DimensionsBasis Drawing Vectors

Basis A basis is a set of linearly independent vectors which spans a space. This means any point in the space can be written as a combination of scalar multiples of the basis vectors. Next What is a Vector? DimensionsBasis Drawing Vectors

Basis Vectors Lets think about the basis vectors of the 2D space. Any point in this space can be described as the vector. We can also write this in the following way: Next What is a Vector? DimensionsBasis Drawing Vectors

Basis Vectors We say the vectors and are basis vectors of the 2D space. In a similar way,,, are basis vectors of the 3D space. e 1, e 2 and e 3 are also called i, j and k. Next What is a Vector? DimensionsBasis Drawing Vectors

Question…. Which of the following is equivalent to ? What is a Vector? DimensionsBasis Drawing Vectors

( ) Vector in two dimensions x y a b v v = v = i+ j ab Here we find the position relative to the origin Click here to see how the vector is formed. Next What is a Vector? DimensionsBasis Drawing Vectors

x y

x y v a b Click here to Repeat Click here to Go Back

Try drawing some vectors x y v = v = i + j Next What is a Vector? DimensionsBasis Drawing Vectors

x z a b c v = y v v = i+ j+ k abc Vector in three dimensions Click here to see how the vector is formed. Next What is a Vector? DimensionsBasis Drawing Vectors

x z y

x z y v Click to here repeat Click here to go back a b c

Which of the following is equivalent to this vector: x z y v What is a Vector? DimensionsBasis Drawing Vectors

Conclusion Vectors are quantities that have a size and a direction. They can be in any number of dimensions. The number of basis vectors required is equal to the number of dimensions. Next What is a Vector? DimensionsBasis Drawing Vectors