Vectors

This section only expects students to have familiarity with measuring distance and coordinates in three dimensions. If your students are very familiar with vectors and their basic operations, then this section can be covered in one class day. Some instructors present the notation and definitions of vectors and associated operations without a lot of motivation. While this algebraic oriented approach can save some time in coverage now, the tradeoff is that students often need more support with how to view vectors geometrically (often in the next several sections).

In this text, we have opted to use the over-arrow notation for vectors since that most easily translates to the way that students can write a clear notation in handwriting. We do not put a precedence on using ijk notation versus component form but will often switch between notations to familiarize students with both forms, as they will likely encounter both of these notations in future courses and other sources.

offers a basic way for students to familiarize themselves with position vectors as a tool to describe locations. offers motivation for why you would want to add, subtract, and rescale vectors using a navigation/mapping motivation. is a great preview of our development of lines in space and uses of vectors in standard position. The other activities in this section are calculation and algebraically oriented.

Most of the exercises in this section are also oriented toward calculations and algebraic forms used in vector problems. offers a basic form diagram problem that many students will see in physics courses. is an activity that some instructors may skip over but offers a nice connection between the algebra of linear combinations and the construction of the vector equation of a line.

Quantities like length, speed, area, and mass are all measured using a single number (a scalar). Other quantities, like velocity, force, and displacement, have two attributes: magnitude (or length) and direction. These quantities are represented by vectors and are the focus of this section. For example, we will use vectors to calculate work done by a constant force, calculate torque, determine direction vectors for lines and normal vectors for planes, define curvature, and determine the direction of greatest increase on a surface. For most of these applications, we will be interested in using vectors to measure the direction of some aspect of the application while needing to also pay attention to the magnitude of this measurement. Vectors will be an essential tool for describing the behavior of functions of several variables in Chapters 11 through 13.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

shows how we can record the length and direction of a change in position using a pair of numbers \(\langle x,y\rangle\). In particular, these numbers measure the change in each coordinate direction separately. There are many quantities other than displacement, such as force and velocity, that possess the attributes of magnitude and direction. We will call such quantity a vector.

This definition is a common geometric idea but some students will have seen an introduction to linear algebra at this point. In linear algebra, a vector is any kind of object that satisfies the definition of an element of a vector space (see the Wikipedia definition of a vector space). The idea of a vector can include A LOT of objects that do not look like arrows in two or three dimensions, but the ideas, like linear combinations and measuring length of a vector, are central to why linear algebra is a hugely impactful branch of mathematics.

We can represent a vector geometrically as a directed line segment, with the magnitude as the length of the segment and an arrowhead indicating direction, as shown at left in .

According to the definition, a vector possesses the attributes of magnitude (length) and direction; the vector's position, however, is not mentioned. Consequently, any two vectors having the same magnitude and direction are equal, as shown in . In other words, two vectors are equal provided they have the same magnitude and direction. This means that the same vector may be drawn in the plane in many different ways.

Vectors do not have a particular starting or ending point. However, it will be useful to use directed line segments like \(\overrightarrow{AB}\) as a specific representative of a vector. For instance, suppose that we would like to draw the vector \(\langle 3, 4\rangle\), which represents a horizontal change of (positive) three units and a vertical change of (positive) four units. We may place the tail of the vector (the point from which the vector originates) at the origin and the tip (the terminal point of the vector) at \((3,4)\), as illustrated in Figure. A vector with its tail at the origin is said to be in standard position.

The point at the tail of a vector is often called the initial point or starting point. The point at the end of the vector is often called the terminal point or final point. The entries of the vector measure the change in each coordinate separately. Remember that change is measured as final minus initial, in general.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

Because initial and final locations are not mentioned in the definition of a vector, any two vectors that have the same magnitude and direction are equal. It is helpful to have an algebraic way to determine when this occurs. That is, if we know the components of two vectors \(\vu\) and \(\vv\), we will want to be able to determine algebraically when \(\vu\) and \(\vv\) are equal. There is an straightforward set of conditions that we use in terms of components.

Two vectors \(\vu = \langle u_1, u_2 \rangle\) and \(\vv = \langle v_1, v_2 \rangle\) in \(\R^2\) are equal if and only if their corresponding components are equal: \(u_1 = v_1\) and \(u_2 = v_2\).

More generally, two vectors \(\vu = \langle u_1, u_2, \ldots, u_n\rangle\) and \(\vv = \langle v_1, v_2, \ldots, v_n \rangle\) in \(\R^n\) are equal if and only if \(u_i = v_i\) for every \(i\) from \(1\) to \(n\).

Vectors are not numbers (scalars), but we can now represent vectors with components that are real numbers. As such, we naturally wonder if it is possible to add two vectors together, multiply two vectors, or combine vectors in any other ways. In this section, we will study two operations on vectors: vector addition and scalar multiplication. To begin, we investigate a natural way to think about combining vectors in the context of measuring the change of location.

is a conceptual activity that motivates the addition, the subtraction, and the rescaling of vectors and we recommend that you do this with a graph or grid paper for students to label and plot the locations and associated vectors.

In we saw how adding two vectors componentwise gave a vector that represented the total change of traveling along one vector followed by doing the change described by the other vector. Additionally, we saw that to go three times as far in a direction given by a vector, we can describe the total change by multiplying each component by three. Finally, we saw that we can find the difference between two vectors by componentwise subtraction. This difference vector completed a triangle formed by positioning the two subtracted vectors with the same initial point. The ideas of adding vectors or multiplying a scalar by a vector generalizes beyond the context of .

In general, we can add vectors \(\vv=\langle v_1,\ldots,v_n \rangle\) and \(\vw=\langle w_1,\ldots,w_n \rangle\) componentwise to get \(\vv+\vw=\langle v_1+w_1,\ldots,v_n+w_n\rangle\), where \(\vv+\vw\) gives the total change described by \(\vv\) then \(\vw\). We define the scalar multiplication of vectors componentwise as well. If \(k\) is a real number, then \(k\vv = \langle k v_1,\ldots, k v_n \rangle\). Geometrically, the scalar multiplication of \(k\) by \(\vv\) will scale (stretch/shrink) \(\vv\) by a factor of \(k\). We will discuss the way different values of \(k\) scale vectors in .

We can now add vectors and multiply vectors by scalars, and thus we can add together scalar multiples of vectors. This allows us to define vector subtraction, \(\vv - \vu\), as the sum of \(\vv\) and \(-1\) times \(\vu\), so that \[\begin{aligned}\end{aligned}\] You can alternatively think of of \(\vv-\vu\) as the vector you need to add to \(\vu\) to get a result of \(\vv\) (as you did in ).

In \(\R^3\), we let \(\vi = \langle 1, 0, 0 \rangle\), \(\vj = \langle 0,1,0 \rangle\), and \(\vk = \langle 0,0,1 \rangle\), and we can write the vector \(\langle a, b, c \rangle\) in \(\R^3\) as \[\begin{aligned}\end{aligned}\]

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

We know that the sum (of scalars) \(1+2\) is equal to the sum \(2+1\). This is called the commutative property of scalar addition. Any time we define operations on objects (like addition of vectors) we usually want to know what kinds of properties the operations have. For example, is addition of vectors a commutative operation? To answer this question we take two arbitrary vectors \(\vv\) and \(\vu\) and add them together and see what happens. Let \(\vv = \langle v_1, v_2 \rangle\) and \(\vu = \langle u_1, u_2 \rangle\). Now we use the fact that \(v_1\), \(v_2\), \(u_1\), and \(u_2\) are scalars, and that the addition of scalars is commutative to see that \[\begin{aligned}\vv + \vu =\mathstrut \amp \langle v_1, v_2 \rangle + \langle u_1, u_2 \rangle \\ =\mathstrut \amp \langle v_1+u_1, v_2 + u_2 \rangle \\ =\mathstrut \amp \langle u_1+v_1, u_2+v_2 \rangle \\ =\mathstrut \amp \langle u_1, u_2 \rangle + \langle v_1, v_2 \rangle \\ =\mathstrut \amp \vu + \vv.\end{aligned}\]

So the vector sum is a commutative operation. Similar arguments can be used to show the following properties of vector addition and scalar multiplication.

Let \(\vv\), \(\vu\), and \(\vw\) be vectors in \(\R^n\) and let \(a\) and \(b\) be scalars. Then

1. \(\vv + \vu = \vu + \vv\)

2. \((\vv + \vu) + \vw = \vv + (\vu + \vw)\)

3. The vector \(\vzero = \langle 0, 0, \ldots, 0 \rangle\) has the property that \(\vv + \vzero = \vv\). The vector \(\vzero\) is called the zero vector.

4. \((-1)\vv + \vv = \vzero\). The vector \((-1)\vv = -\vv\) is called the additive inverse of the vector \(\vv\).

5. \((a+b) \vv = a\vv + b\vv\)

6. \(a(\vv + \vu) = a\vv + a\vu\)

7. \((ab) \vv = a(b\vv)\)

8. \(1 \vv = \vv\).

We verified the first property for vectors in \(\R^2\). The other properties can be verified to hold for all vectors in \(\R^n\) in a similar manner because the operations are done componentwise on the vectors.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

Next, we explore the geometric representations of vector addition and scalar multiplication which will allow us to visualize these operations. Let \(\vu = \langle 4, 6 \rangle\) and \(\vv = \langle 3, -2 \rangle\). Then \(\vw = \vu + \vv = \langle 7, 4 \rangle\), as shown in .

If we think of these vectors as displacements measuring change in position in the plane, we can see a geometric way to envision vector addition. For instance, the vector \(\vu + \vv\) will represent the displacement obtained by following the displacement \(\vu\) with the displacement \(\vv\). We may picture this by placing the tail of \(\vv\) at the tip of \(\vu\), as seen in . In other words, the change described by \(\vu+\vv\) is given by going along \(\vu\) then immediately along \(\vv\).

Vector addition is commutative. This means we obtain the same sum if we place the tail of \(\vu\) at the tip of \(\vv\) as if we place the tail of \(\vv\) at the tip of \(\vu\). We therefore see that \(\vu+\vv\) appears as the diagonal of the parallelogram determined by \(\vu\) and \(\vv\), as shown in .

Vector subtraction has a similar interpretation. In we see vectors \(\vu\), \(\vv\), and \(\vw =\vu + \vv\). If we rewrite \(\vv = \vw - \vu\), we have the arrangement shown at right in . In other words, to form the difference \(\vw-\vu\), we draw a vector from the tip of \(\vu\) to the tip of \(\vw\). This is consistent with our other interpretation of vector subtraction as well: the vector \(\vw-\vu\) is what we need to add to \(\vu\) to get \(\vw\). This is stated algebraically as \(\vu + \underline{\vw-\vu} = \vw\).

In a similar way, we may geometrically represent a scalar multiple of a vector. For instance, if \(\vv=\langle 2,3\rangle\), then \(2\vv = \langle 4,6\rangle\). As shown in Figure, multiplying \(\vv\) by 2 leaves the direction unchanged, but stretches \(\vv\) by 2. Remember that \(2\vv = \vv +\vv\), which describes the total change of doing \(\vv\) then \(\vv\) again and thus gives a result that will be in same direction as \(\vv\). Also, \(-2\vv = \langle -4, -6\rangle\), which shows that multiplying by a negative scalar gives a vector pointing in the opposite direction of \(\vv\).

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

By definition, vectors have both direction and magnitude (or length). We will now investigate how to calculate the magnitude of a vector. Since a vector \(\vv\) can be represented by a directed line segment, we can use the distance formula to calculate the length of the segment. This length is the magnitude of the vector \(\vv\) and is denoted \(\vecmag{\vv}\).

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.