Vectors and Scalars - Part I

Some quantities are expressed as Vectors and others are expressed as Scalars. Since Vectors and Scalars are fundamental concepts that are used in Engineering and Physics, I thought I might spend a blog entry or two on their features. Let's start off with some informal definitions.

Scalar- A magnitude only (Energy, Temperature, Voltage, etc.)

Vector- A magnitude and a direction (Force, Current, Velocity, etc.)

Scalars and Vectors can and will interact with each other. It is useful to know each of the interactions:

Addition

Scalar + Scalar = Scalar (Always)

Scalar + Vector = Undefined

Vector + Vector = Vector (Always)

Multiplication

Scalar x Scalar = Scalar (Always)

Scalar x Vector = Vector (Always)

Vector x Vector = Scalar (Commonly called the Dot Product)

Vector x Vector = Vector (Commonly called the Cross Product)

So you can add vectors, take the dot product or cross product of vectors, or multiply vectors by a scalar. You can add and multiply scalars.

Just as scalars can be plotted on a graph, so can vectors. The most intuitive expression of vectors is in Spherical Coordinates (ρ,Φ,θ), where ρ is equal to the magnitude of the vector and Φ and θ coordinates are used to indicate its direction. Unfortunately Cartesian Coordiantes (x,y,z) aren't as convenient because the magnitude and direction aren't split up betwen x, y, and z but all mixed together.

Since this mixing in x,y,z coordinates can be a liability, it is usefull to be able to switch between x,y,z coordinate systems and ρ,Φ,θ coordinate systems. Basically:

Since much of the time we are more interested in the magnitude of the vector as opposed to its direction, we are only interested in ρ. Sometimes r is used instead of ρ. This is why many equations are written in the form similiar to the coulomb force below:

In the equation above, the r is the magnitude of the distance between the two charges q₁ and q₂ and is the direction, called the unit vector. All the unit vector is is the direction with a magnitude of 1. The equation is splitting the information up into magnitude and direction because force is a vector.

Another feature of vectors is they can used to form vector fields. To explain this, lets first consider a scalar field and then extend the idea to vectors.

A scalar field assigns a magnitude for every point in space. So if the coordinates are in the form of (x,y,z) then for each unique value of (x,y,z) there is an associated scalar, such as energy or temperature. Lets take the temperature of a room as the scalar we're interested in. If we take the temperature of each point in the room and write it down, we have basically written down the temperature scalar field of the room.

Now remember vectors are the same as scalars except each point would have a magnitude and direction. If we were to measure the wind velocity in a room for every point (x,y,z), we'd have to make a list that had three columns, the point in the room where we took the measurement, velocity magnitude, velocity direction. If we made this list for every point in the room, we have basically written down the Wind Velocity Vector field of the room. The image below demonstrates what a vector field looks like:

Sometimes (like in the graph above) a scalar or vector field can be described by an equation. This equation is called a function and can be used to calculate the vector or scalar, depending on whether its a vector or scalar function, for every point in space. In my next blog I will spend a little time talking about vector functions and then I will revisit an earlier blog topic, namely the gradient, curl, and divergence of a field.