Scalars and vectors are used in science to mathematically describes certain quantities (e.g. temperature, or velocity) in order to know what sort of mathematical operations can be properly done on them to give real results. The important phrase is “mathematically describe” – if we do not understand that quantity mathematically, then we have no idea how to use that quantity in our formulas and derivations.
Understanding scalar and vector quantities are simple. Scalar quantities are defined by only one number. For example, Temperature is a scalar quantity – you only need one number to describe temperature. Vector quantities are defined by two numbers. For example, Velocity is a vector quantity – you need two numbers to fully describe it – magnitude (how large it’s value is), and an angle or a set of coordinates( to show in which direction it is pointed). Let’s say you decide to drop the angle – now you have Speed, which is defined by only one number, the magnitude. Speed is a scalar quantity. Other examples of scalars are mass, energy, and charge.
So far we have only described scalars and vectors as quantities, and for temperature and velocity they are enough. But we can get more complicated by creating a formula that produces a scalar or a vector depending on what set of points (for example, coordinates of an axis system) you put in. That formula is called a “field”, and you can create scalar fields, and vector fields.
Let’s look at an example of a scalar field. Let’s use your bedroom floor and put an x-y axis system to describe every point on your floor by the coordinate pair (x,y). For example, (3,4) would be three up and four across starting from one corner of your floor. Let’s also say we have a formula called T(x,y) that will give you the temperature of you bedroom floor a point (x,y). We can now say that T(x,y) is a two dimensional SCALAR FIELD – it takes the input of (x,y) – a point on your floor – and gives you ONE number – the temperature of the floor at that point. Another way to say it is T(x,y) is a FIELD of scalars. If we wanted to include every point in your room – i.e. go three dimensional and use coordinates x,y, and z – you just need a formula T(x,y,z) so that when you put in a point (x,y,z) you still get out ONE number – the temperature at the point. T(x,y,z) is now a three-dimensional scalar field. You just need to put in 3-D coordinate point to get a scalar quantity back.
Now let’s look at an example of a vector field. Let’s use your floor again and use an x-y coordinate system to define all the points on the floor. This time we want to know the velocity of the air – how fast and which direction it’s going – given an (x,y) point. We create a formula, V(x,y), a two-dimensional VECTOR field – you put in two numbers, x and y, and you get out TWO numbers, magnitude and direction of the velocity of air at x and y. Thus, V(x,y) is a FIELD of vectors. Think of a flat sheet with arrow pointing willy-nilly like a row of crops growing in weird directions. We can extend this to three dimensions, V(x,y,z), and now we have to put in an x,y,z coordinate to get back a vector quantity. Thus, V(x,y,z) is a three-dimensional vector field.
There are other examples of vector fields. In electromagnetism we have electric FIELDS – i.e. a formula where we put in a coordinate and we get back the magnitude and direction of that electric field. Thus, electric fields are vector fields. Likewise for gravitational fields and so on.
Since we have looked at fields, we can now look at scalars and vectors as defined by a particular formula. When we do this we have to bring in additional mathematical constraints to define scalars and vectors. Why? Because we can use many coordinate systems – for example, Cartesian (x,y) or Polar (r, theta) – and the formulas we use better give us the correct (not necessarily the same) answe for whichever coordinate system we use. (Physical phenomena should not depend on which coordinate system we use.) Thus we say a field is a scalar field when it’s value is the same regardless of any coordinate transformation. There are examples of pseudo-scalars where the values changes sign under certain transformations. We say a field is a vector field when it give the correct answer under proper or improper rotations (i.e. transformations that rotate and cause a sign flip). If the vector field does not give the correct answer under improper rotations, then that formula is a pseudo-vector field.
We can use this concept to go back and define pseudo-scalar and pseudo-vector quantities. Examples of psuedovectors are angular momentum or torque. Examples of psuedoscalars are the elementary particles pions and mesons, since under parity transformations, their charge signs flip.
You now have a wordy understanding and a set of examples of scalars and vectors as both quantities and fields, and hopefully you can see how these objects can be used in science so that various mathematical tools can be employed upon them to develop new solutions to physic questions. To have a proper feel for them however you will need to understand and solve several physics problems to know all the mathematical technique used to manipulate scalar and vectors. One good source in Mathematical Methods by Mary Boas, or for an advanced student (graduate level) Mathematical methods by Arfken.