Chapter 2: Vectors

2.0. Overview

Vectors are quantities that have both a magnitude and a direction. Quantities that can be expressed as vectors include velocity, acceleration, displacement, force, momentum, electric field, magnetic field,...

We'll be using vectors throughout this course, so having a solid grasp of the fundamentals is critical.

Let's go!

1. Vector Definition
2. Graphical Analysis
3. Adding and Subtracting Vectors
4. Polar Notation of Vectors
5. Unit Vectors (i, j)

2.1. Vector Definition

Some measurements—scalar quantities—are indicated by simply a number (and a unit): the height of a playing card (in inches? or centimeters?), the mass of penny (in grams, perhaps), the age of a person (in years, possibly).

But some measurements are vectors. Vectors have two characteristics: magnitude (the numeric part of the measurement), and direction (the orientation of the vector); and we’ve talked about three kinds of vector quantities so far:

• displacement x
• velocity v
• acceleration a

Up till now, we’ve considered very simple straight line motion, restricting ourselves to one-dimensional horizontal motion (in the +x or -x direction), or vertical motion (in the +y or -y direction). We need to expand our abilities now into additional dimensions.

2.2. Graphical Analysis

One way of analyzing 2-dimensional situations with vectors is by drawing the vector on paper, according to the following rules.

• the length of a line drawn is proportional to the magnitude of the vector quantity
• the direction that the arrow points (relative to an Cartesian coordinate system) indicates the direction of vector quantity

The following diagrams illustrate vectors that indicate various quantities.

2.3. Adding and Subtracting Vectors

It turns out that there are lots of things we can use vectors for: adding vectors, for example, allows us to determine the combined effects of two different motions. A plane flying north in air that is being blown to the west, for example, will have a direction relative to the earth that is not directly north.

Throughout this course we will be drawing diagrams to represent situations, and this includes working with vectors.

2.3.1. Adding vectors graphically

Adding two or more vectors graphically is accomplished by drawing the vectors such that the head (the arrow) of one vector is touching the tail of the next vector, in any order. The net, or resultant vector is found by drawing a line from the tail of the first vector to the head of the last vector.

If we were to add a vector A to a vector B to form a vector C, for example, it would be written as . We would sketch the diagram by first drawing vector A (here it is oriented to the northeast somewhat), and then draw the tail of the vector B (which represents some sort of vertical quantity) starting at the tip/head/arrow of A. Finally we would draw the resultant C vector by sketching a line from the beginning of A to the end of B. This final vector represents the net, or resultant, vector.

To identify the magnitude and direction of the resultant C quantity, there are a number of ways you could determine those values. If solving graphically, you'd take a ruler and a protractor and measure the vector on your drawing.

This type of determination isn't as precise as most people would like, but drawing the sketch and being able to make a rough determination can be useful.

2.3.2. Vector Subtraction

How would one go about graphically subtracting one vector from another?

In the same way that A - B is equal to A + (-B), we can take the negative of a vector—a vector of the same magnitude but pointed in the opposite direction—and add it (place it tip-to-tail) with the first vector.

2.2.3. Specifying Direction

When giving the directions for a vector, it's best to be as clear as possible. Just saying "generally northeast" isn't as good as saying "36.9°," where that angle is typically specified relative to the positive-x axis.

Sometimes a problem will specify "20° North of West," and you can use that to figure out what's going on. In specifying your own answers, however, you should stick to using angles based on a unit circle, with 0 on the positive x-axis, and measuring counterclockwise from there. For small angles, it's okay to give answers in negative degrees as well.

Obviously, graphical addition of vectors is fine for sketches, but measuring the resultant vector with a ruler and protractor isn't very precise when it comes to numeric solutions. For that, we have two slightly more complex, but incredibly useful, systems: the polar-notation system, and the unit-vector system.

2.4. Polar Notation of Vectors

Using polar notation involves describing vectors in terms of their magnitude and direction, as we have already done, eg. 10 m/s @ 36.9°. To be able to perform vector addition we're going to need to be able to split a vector up into its x and y components, which requires using some simple trigonometry.

How, specifically, do we do that?

2.4.1. Adding Polar Notation Vectors

Once we're comfortable with the idea of considering a vector in terms of its x-y components, we can take advantage of a powerful fact:

The sum of the vector components in any direction is equal to the resultant component in that direction.

Let's see how that works.

2.5. Unit Vectors (i, j)

Yet another way exists to designate, and calculate with, vectors: unit vectors.

There's not much added value in being able to indicate one-dimensional motion using unit vectors, which is why you may not have learned about them before now. But now that we're working in 2-dimensional or 3-dimensional motion, and now that we know we need to break vectors up into components on a regular basis, it's very convenient to be able to say that something's velocity is "3î + 4ĵ m/s."

You can certainly deduce that the overall (net) velocity of such an object is 5 m/s, and you can calculate tan^-1(4 / 3) to identify the direction (angle) as 53.1°. That's important to know.

But if we have to do any analyses that require us breaking this vector up into x and y components, it already has been! The "5 m/s at 53.1°" consists of two components, one in the x direction and one in the y direction. And we can use those components to solve problems.

Now that we know a little about unit vectors, let's try another problem.

2.5.1. Derivatives of Unit Vectors (i, j)

We typically use x to indicate a position along the x-axis, and y for the y-position. If we want to indicate a position based on multiple axes coordinates, we often use r to indicate that position.

If you were asked to determine Δx, the displacement between two positions x_final and x_initial, that wouldn't be difficult to do.

Likewise, determine the displacement between two r-vector positions in space is fairly straightforward.

Another classic problem is to determine the time-derivative of one of these unit-vector expressions. Consider this problem.