Vectors Definition: A vector quantity is one which has both magnitude and direction One of the simplest vectors is called a displacement… it has an associated distance (magnitude) and direction. For example, if we have two points A and B then ‘displacement AB’ is a vector quantity. It can be conveniently represented by a straight line joining A and B, and the direction indicated by an arrow. A B

The length of the line AB represents the distance between A and B The length of the line AB represents the distance between A and B. For ‘vector AB’ we write either AB or and textbooks use bold type. Another notation involves the use of lower case letters ‘vector r’ r

Definition: A scalar quantity is one which has magnitude only. In the context of the previous example ‘distance travelled’ is a scalar quantity. A displacement vector in two dimensions is conveniently expressed in component form, as a column vector. a b c d

a + b a – b Addition and Subtraction using components We can see displacement a followed by displacement b which is… c b entirely equivalent to the single displacement c. We write c = a + b a a + b a – b

Multiplication of a vector by a scalar Consider the vectors a, 3a, -a, -2a a 3a -a -2a They are parallel vectors Parallel vectors If a and b are parallel then for some scalar quantity, , it must follow that a = b If a = b then a and b are parallel

Modulus of a vector: is its magnitude and hence a scalar. The modulus of a is written |a| and is written usually as a. Magnitude of a vector in component form Example a In general