Vector Notation

Vectors are used in aerial navigation.

The idea of components freed us from the confines of one-dimensional physics, but the component notation can be unwieldy, since every onedimensional equation has to be written as a set of three separate equations in the three-dimensional case. Newton was stuck with the component notation until the day he died, but eventually someone sufficiently lazy and clever figured out a way of abbreviating three equations as one.

Example (a) shows both ways of writing Newton's third law. Which would you rather write?

The idea is that each of the algebra symbols with an arrow written on top, called a vector, is actually an abbreviation for three different numbers, the x, y, and z components. The three components are referred to as the components of the vector, e.g., F_x is the x component of the vector

The notation with an arrow on top is good for handwritten equations, but is unattractive in a printed book, so books use boldface, F, to represent vectors. After this point, I'll use boldface for vectors throughout this book.

In general, the vector notation is useful for any quantity that has both an amount and a direction in space. Even when you are not going to write any actual vector notation, the concept itself is a useful one. We say that force and velocity, for example, are vectors. A quantity that has no direction in space, such as mass or time, is called a scalar. The amount of a vector quantity is called its magnitude. The notation for the magnitude of a vector A is |A|, like the absolute value sign used with scalars.

Often, as in example (b), we wish to use the vector notation to represent adding up all the x components to get a total x component, etc. The plus sign is used between two vectors to indicate this type of componentby- component addition. Of course, vectors are really triplets of numbers, not numbers, so this is not the same as the use of the plus sign with individual numbers. But since we don't want to have to invent new words and symbols for this operation on vectors, we use the same old plus sign, and the same old addition-related words like "add," "sum," and "total." Combining vectors this way is called vector addition.

Similarly, the minus sign in example (a) was used to indicate negating each of the vector's three components individually. The equals sign is used to mean that all three components of the vector on the left side of an equation are the same as the corresponding components on the right.

Example (c) shows how we abuse the division symbol in a similar manner. When we write the vector Δv divided by the scalar Δt, we mean the new vector formed by dividing each one of the velocity components by Δt.

It's not hard to imagine a variety of operations that would combine vectors with vectors or vectors with scalars, but only four of them are required in order to express Newton's laws:

operation	definition
vector + vector	Add component by component to make a new set of three numbers.
vector - vector	Subtract component by component to make a new set of three numbers.
vector · scalar	Multiply each component of the vector by the scalar.
vector/scalar	Divide each component of the vector by the scalar.

As an example of an operation that is not useful for physics, there just aren't any useful physics applications for dividing a vector by another vector component by component. In optional section 7.5, we discuss in more detail the fundamental reasons why some vector operations are useful and others useless.

We can do algebra with vectors, or with a mixture of vectors and scalars in the same equation. Basically all the normal rules of algebra apply, but if you're not sure if a certain step is valid, you should simply translate it into three component-based equations and see if it works.

Order of addition.

It is useful to define a symbol r for the vector whose components are x, y, and z, and a symbol Δr made out of Δx, Δy, and Δz.

Although this may all seem a little formidable, keep in mind that it amounts to nothing more than a way of abbreviating equations! Also, to keep things from getting too confusing the remainder of this chapter focuses mainly on the Δr vector, which is relatively easy to visualize.

Self-Check	Translate the equations vx = Δx/Δt, vy = Δy/Δt, and vz = Δz/Δt for motion with constant velocity into a single equation in vector notation.
Answer	v = Δr/Δt

Drawing vectors as arrows

a / The x an y components of a vector can be thought of as the shadows it casts onto the x and y axes.

A vector in two dimensions can be easily visualized by drawing an arrow whose length represents its magnitude and whose direction represents its direction. The x component of a vector can then be visualized as the length of the shadow it would cast in a beam of light projected onto the x axis, and similarly for the y component. Shadows with arrowheads pointing back against the direction of the positive axis correspond to negative components.

In this type of diagram, the negative of a vector is the vector with the same magnitude but in the opposite direction. Multiplying a vector by a scalar is represented by lengthening the arrow by that factor, and similarly for division.

Self-Check	Given vector Q represented by an arrow below, draw arrows representing the vectors 1.5Q and -Q.
Answer

Discussion Questions

A

Would it make sense to define a zero vector? Discuss what the zero vector's components, magnitude, and direction would be; are there any issues here? If you wanted to disqualify such a thing from being a vector, consider whether the system of vectors would be complete. For comparison, can you think of a simple arithmetic problem with ordinary numbers where you need zero as the result? Does the same reasoning apply to vectors, or not?

B

You drive to your friend's house. How does the magnitude of your Δr vector compare with the distance you've added to the car's odometer?