Why learn about vectors?

It would be hard to overstate the importance of vectors for introductory-level physics. Many of the physical quantities that describe how and why an object moves, like velocity, acceleration, and force, are vector quantities. In addition, we will return time again to vector problems as we go from working Newton’s Second Law problems to Conservation of Momentum problems to problems solving for the net field at a point in space. Obviously, if we are going to spend a significant fraction of the course working with vectors, we would like to be as comfortable as possible with the mathematical processes we will use with vectors. The more comfortable we are with our vector operations, the easier it will be to see what the vectors are trying to tell us about the underlying physics in a particular situation.

The goal of this first set of modules is to introduce the five crucial skills that we need to develop when working with vectors:

1. Breaking vectors up into components if we know their magnitudes and directions,
2. Calculating the magnitude and direction of a vector if we know its components,
3. Adding two or more vectors together,
4. Subtracting two or more vectors, and
5. Multiplying a vector by a scalar to generate a new vector.

You can think of this set of skills as the important tools in your toolbox when we are working vector problems. We want to be proficient at both selecting and using each tool as we work through problems.

The first skill, breaking vectors up into their components, is crucial. Many of the mistakes students make in adding and subtracting vectors are actually errors in determining the x and y components of the vectors in the problem. Fortunately, using some of the fundamental relations from trigonometry, there is a relatively straight-forward way to calculate the components of a vector given its magnitude and direction. Therefore the first module will start with a brief review of the relations from trigonometry that we will find useful before describing what vectors are and how we will determine their components.