Many quantities we think about daily can be described by a single number: temperature, speed, cost, weight and height. There are also many other concepts we encounter daily that cannot be described with just one number. For instance, a weather forecaster often describes wind with its speed and its direction (“\(\ldots\) with winds from the southeast gusting up to 30 mph \(\ldots\)”). When applying a force, we are concerned with both the magnitude and direction of that force. In both of these examples, direction is important. Because of this, we study vectors, mathematical objects that convey both magnitude and direction information.

One “bare–bones” definition of a vector is based on what we wrote above: “a vector is a mathematical object with magnitude and direction parameters.” This definition leaves much to be desired, as it gives no indication as to how such an object is to be used. Several other definitions exist; we choose here a definition rooted in a geometric visualization of vectors. It is very simplistic but readily permits further investigation.

Figure 10.2.2 shows multiple instances of the same vector. Each directed line segment has the same direction and length (magnitude), hence each is the same vector.

We use \(\mathbb{R}^2\) (pronounced “r two”) to represent all the vectors in the plane, and use \(\mathbb{R}^3\) (pronounced “r three”) to represent all the vectors in space.

Consider the vectors \(\vv{PQ}\) and \(\vv{RS}\) as shown in Figure 10.2.3. The vectors look to be equal; that is, they seem to have the same length and direction. Indeed, they are. Both vectors move 2 units to the right and 1 unit up from the initial point to reach the terminal point. One can analyze this movement to measure the magnitude of the vector, and the movement itself gives direction information (one could also measure the slope of the line passing through \(P\) and \(Q\) or \(R\) and \(S\)). Since they have the same length and direction, these two vectors are equal.

This demonstrates that inherently all we care about is displacement; that is, how far in the \(x\text{,}\) \(y\) and possibly \(z\) directions the terminal point is from the initial point. Both the vectors \(\vv{PQ}\) and \(\vv{RS}\) in Figure 10.2.3 have an \(x\)-displacement of 2 and a \(y\)-displacement of 1. This suggests a standard way of describing vectors in the plane. A vector whose \(x\)-displacement is \(a\) and whose \(y\)-displacement is \(b\) will have terminal point \((a,b)\) when the initial point is the origin, \((0,0)\text{.}\) This leads us to a definition of a standard and concise way of referring to vectors.

It follows from the definition that the component form of the vector \(\vv{PQ}\text{,}\) where \(P=(x_1,y_1)\) and \(Q=(x_2,y_2)\) is \begin{equation*} \vv{PQ} = \la x_2-x_1, y_2-y_1\ra; \end{equation*} in space, where \(P=(x_1,y_1,z_1)\) and \(Q=(x_2,y_2,z_2)\text{,}\) the component form of \(\vv{PQ}\) is \begin{equation*} \vv{PQ} = \la x_2-x_1, y_2-y_1,z_2-z_1\ra. \end{equation*}

We practice using this notation in the following example.

Now that we have defined vectors, and have created a nice notation by which to describe them, we start considering how vectors interact with each other. That is, we define an algebra on vectors.

In short, we say addition and scalar multiplication are computed “component–wise.”

As vectors convey magnitude and direction information, the sum of vectors also convey length and magnitude information. Adding \(\vec u+\vec v\) suggests the following idea:

This idea is sketched in Figure 10.2.12, where the initial point of \(\vec v\) is the terminal point of \(\vec u\text{.}\) This is known as the “Head to Tail Rule” of adding vectors. Vector addition is very important. For instance, if the vectors \(\vec u\) and \(\vec v\) represent forces acting on a body, the sum \(\vec u+\vec v\) gives the resulting force. Because of various physical applications of vector addition, the sum \(\vec u+\vec v\) is often referred to as the resultant vector, or just the “resultant.”

Analytically, it is easy to see that \(\vec u+\vec v = \vec v+\vec u\text{.}\) Figure 10.2.12 also gives a graphical representation of this, using gray vectors. Note that the vectors \(\vec u\) and \(\vec v\text{,}\) when arranged as in the figure, form a parallelogram. Because of this, the Head to Tail Rule is also known as the Parallelogram Law: the vector \(\vec u+\vec v\) is defined by forming the parallelogram defined by the vectors \(\vec u\) and \(\vec v\text{;}\) the initial point of \(\vec u+\vec v\) is the common initial point of parallelogram, and the terminal point of the sum is the common terminal point of the parallelogram.

While not illustrated here, the Head to Tail Rule and Parallelogram Law hold for vectors in \(\mathbb{R}^3\) as well.

It follows from the properties of the real numbers and Definition 10.2.9 that \begin{equation*} \vec u-\vec v = \vec u + (-1)\vec v. \end{equation*}

The Parallelogram Law gives us a good way to visualize this subtraction. We demonstrate this in the following example.

The zero vector is the vector whose initial point is also its terminal point. It is denoted by \(\vec 0\text{.}\) Its component form, in \(\mathbb{R}^2\text{,}\) is \(\la 0,0\ra\text{;}\) in \(\mathbb{R}^3\text{,}\) it is \(\la 0,0,0\ra\text{.}\) Usually the context makes is clear whether \(\vec 0\) is referring to a vector in the plane or in space.

Our examples have illustrated key principles in vector algebra: how to add and subtract vectors and how to multiply vectors by a scalar. The following theorem states formally the properties of these operations.

As stated before, each vector \(\vec v\) conveys magnitude and direction information. We have a method of extracting the magnitude, which we write as \(\norm{\vec v}\text{.}\) Unit vectors are a way of extracting just the direction information from a vector.

Consider this scenario: you are given a vector \(\vec v\) and are told to create a vector of length 10 in the direction of \(\vec v\text{.}\) How does one do that? If we knew that \(\vec u\) was the unit vector in the direction of \(\vec v\text{,}\) the answer would be easy: \(10\vec u\text{.}\) So how do we find \(\vec u\) ?

Property 8 of Theorem 10.2.17 holds the key. If we divide \(\vec v\) by its magnitude, it becomes a vector of length 1. Consider: \begin{align*} \snorm{\frac{1}{\norm{\vec v}}\vec v} \amp = \frac{1}{\norm{\vec v}}\norm{\vec v} \amp \text{ (we can pull out \(\ds \frac{1}{\norm{\vec v}}\) as it is a scalar)}\\ \amp = 1. \end{align*}

So the vector of length 10 in the direction of \(\vec v\) is \(\ds 10\frac{1}{\norm{\vec v}}\vec v\text{.}\) An example will make this more clear.

The basic formation of the unit vector \(\vec u\) in the direction of a vector \(\vec v\) leads to a interesting equation. It is: \begin{equation*} \vec v = \norm{\vec v}\frac{1}{\norm{\vec v}}\vec v. \end{equation*}

We rewrite the equation with parentheses to make a point: \begin{equation*} \vec v = \underbrace{\norm{\vec v}}_{\text{ \scriptsize magnitude } }\cdot\underbrace{\left(\frac{1}{\norm{\vec v}}\vec v\right)}_{\text{ \scriptsize direction } }. \end{equation*}

This equation illustrates the fact that a vector has both magnitude and direction, where we view a unit vector as supplying only direction information. Identifying unit vectors with direction allows us to define parallel vectors.

It is equivalent to say that vectors \(\vec v_1\) and \(\vec v_2\) are parallel if there is a scalar \(c\neq 0\) such that \(\vec v_1 = c\vec v_2\) (see marginal note).

If one graphed all unit vectors in \(\mathbb{R}^2\) with the initial point at the origin, then the terminal points would all lie on the unit circle. Based on what we know from trigonometry, we can then say that the component form of all unit vectors in \(\mathbb{R}^2\) is \(\la \cos(\theta) ,\sin(\theta) \ra\) for some angle \(\theta\text{.}\)

A similar construction in \(\mathbb{R}^3\) shows that the terminal points all lie on the unit sphere. These vectors also have a particular component form, but its derivation is not as straightforward as the one for unit vectors in \(\mathbb{R}^2\text{.}\) Important concepts about unit vectors are given in the following Key Idea.

These formulas can come in handy in a variety of situations, especially the formula for unit vectors in the plane.

Unit vectors were very important in the previous calculation; they allowed us to define a vector in the proper direction but with an unknown magnitude. Our computations were then computed component–wise. Because such calculations are often necessary, the standard unit vectors can be useful.

The algebra we have applied to vectors is already demonstrating itself to be very useful. There are two more fundamental operations we can perform with vectors, the dot product and the cross product. The next two sections explore each in turn.