1.1 Vectors in Two- and Three-Dimensional Space

In this section we build our mathematical model of space. Points P in the plane are represented by ordered pairs of real numbers \((a_1,a_2)\); the numbers \(a_1\) and \(a_2\) are called the Cartesian coordinates of P. We draw two perpendicular lines, label them as the \(x\) and \(y\) axes, and then drop perpendiculars from P to these axes, as in Figure 1.1. After designating the intersection of the \(x\) and \(y\) axes as the origin and choosing units on these axes, we produce two signed distances \(a_1\) and \(a_2\) as shown in the figure; \(a_1\) is called the \(x\) component of P, and \(a_2\) is called the \(y\) component.

Points in space may be similarly represented as ordered triples of real numbers. To construct such a representation, we choose three mutually perpendicular lines that meet at a point in space. These lines are called \(x\) axis, \(y\) axis, and \(z\) axis, and the point at which they meet is called the origin (this is our reference point). We choose a scale on these axes, as shown in Figure 1.2.

The triple (\(0,0,0\)) corresponds to the origin of the coordinate system, and the arrows on the axes indicate the positive directions. For example, if \(a,b\) and \(c\) are positive real numbers, then the triple (\(a, b, c\)) represents a point \(a\) units from the origin in the positive direction along the \(x\) axis, \(b\) units in the positive direction along the \(y\) axis, and \(c\) units in the positive direction along the \(z\) axis (Figure 1.3).

Because we can associate points in space with ordered triples in this way, we often use the expression “the point \((a_1,a_2,a_3)\)” instead of the longer phrase “the point P that corresponds to the triple \((a_1,a_2,a_3)\)” We say that \(a_1\) is the \(x\) coordinate (or first coordinate), \(a_2\) is the \(y\) coordinate (or second coordinate), and \(a_3\) is the \(z\) coordinate (or third coordinate) of P. It is also common to denote points in space with the letters \(x,y\), and \(z\) in place of \(a_1,a_2\), and \(a_3\). Thus, the triple \((x,y,z)\) represents a point whose first coordinate is \(x\), second coordinate is \(y\), and third coordinate is \(z\).

By convention, we label the axes as in Figure 1.4, where the positive sides of the axes are labeled \(x\), \(y\), and \(z\). This labeling satisfies the right-hand rule, which means that when you position your right hand so that your fingers curl from the positive \(x\)-axis toward the positive \(y\)-axis, your thumb points in the positive \(z\)-direction. The axes in Figure 1.4 are not labeled according to the right-hand rule.

We employ the following notation for the line, the plane, and three-dimensional space:

• (i) The real number line is denoted \({\mathbb R}^1\) or simply \({\mathbb R}\).
• (ii) The set of all ordered pairs \((x,y)\) of real numbers is denoted \({\mathbb R}^2\).
• (iii) The set of all ordered triples \((x,y,z)\) of real numbers is denoted \({\mathbb R}^3\).

When speaking of \({\mathbb R}^1,{\mathbb R}^2\), and \({\mathbb R}^3\) simultaneously, we write \({\mathbb R}^n\), where \(n=1,2\), or 3; or \({\mathbb R}^m\), where \(m=1,2,3\). Starting in Section 1.5 we will also study \({\mathbb R}^n\) for \(n=4,5,6,\ldots,\) but the cases \(n=1,2,3\) are closest to our geometric intuition and will be stressed throughout the book.

Addition and Scalar Multiplication of Ordered Pairs and Triples

The operation of addition can be extended from \({\mathbb R}\) to \({\mathbb R}^2\) and \({\mathbb R}^3\). For \({\mathbb R}^3\), this is done as follows. Given the two triples \((a_1,a_2,a_3)\) and \((b_1,b_2,b_3)\), we define their sum to be \[ (a_1,a_2,a_3)+(b_1,b_2,b_3)=(a_1+b_1,a_2+b_2,a_3+b_3). \]

The element (\(0, 0, 0\)) is called the zero element (or just zero) of \({\mathbb R}^3\). The element \((-a_1,-a_2,-a_3)\) is the additive inverse (or negative) of \((a_1,a_2,a_3)\), and we will write \((a_1,\) \(a_2\), \(a_3)-(b_1,b_2,b_3)\) for \((a_1,a_2,a_3)+(-b_1,-b_2,-b_3)\).

The additive inverse, when added to the vector itself, of course produces zero: \[ (a_1, a_2, a_3) +(-a_1, -a_2, -a_3) =(0, 0, 0). \]

There are several important product operations that we will define on \({\mathbb R}^3\). One of these, called the inner product, assigns a real number to each pair of elements of \({\mathbb R}^3\). We shall discuss it in detail in Section 1.2. Another product operation for \({\mathbb R}^3\) is called scalar multiplication (the word “scalar” is a synonym for “real number”). This product combines scalars (real numbers) and elements of \({\mathbb R}^3\) (ordered triples) to yield elements of \({\mathbb R}^3\) as follows: Given a scalar \(\alpha\) and a triple \((a_1,a_2,a_3)\), we define the scalar multiple by \[ \alpha (a_1,a_2,a_3) = (\alpha a_1,\alpha a_2,\alpha a_3). \]

Addition and scalar multiplication of triples satisfy the following properties:

• (i) \((\alpha\beta)(a_1, a_2, a_3) = \alpha[\beta(a_1, a_2, a_3)]\)(associativity)
• (ii) \((\alpha + \beta)(a_1, a_2, a_3) = \alpha(a_1, a_2, a_3) + \beta(a_1, a_2, a_3)\)(distributivity)
• (iii) \(\alpha[(a_1, a_2, a_3) + (b_1, b_2, b_3)] = \alpha(a_1, a_2, a_3) + \alpha(b_1, b_2, b_3)\)(distributivity)
• (iv) \(\alpha(0, 0, 0) = (0, 0, 0)\)(property of zero)
• (v) \(0(a_1, a_2, a_3) = (0, 0, 0)\)(property of zero)
• (vi) \(1(a_1, a_2, a_3) = (a_1, a_2, a_3)\)(property of the unit element)

The identities are proven directly from the definitions of addition and scalar multiplication. For instance, \begin{eqnarray*} (\alpha+\beta)(a_1,a_2,a_3) &=& ((\alpha+\beta)a_1,(\alpha+\beta)a_2, (\alpha+\beta)a_3)\\ &=& (\alpha a_1+\beta a_1, \alpha a_2+\beta a_2, \alpha a_3+\beta a_3)\\ &=& \alpha(a_1,a_2,a_3)+\beta(a_1,a_2,a_3). \end{eqnarray*}

For \({\mathbb R}^2\), addition and scalar multiplication are defined just as in \({\mathbb R}^3\), with the third component of each vector dropped off. All the properties (i) to (vi) still hold.

For example, \((1,1)+(2,3)=(3,4)\) and \(2(2,3)=(4,6)\). Also \((0,0)\) is the zero element of \({\mathbb R}^2\).

Geometry of Vector Operations

Let us turn to the geometry of these operations in \({\mathbb R}^2\) and \({\mathbb R}^3\). For the moment, we define a vector to be a directed line segment beginning at the origin; that is, a line segment with specified magnitude and direction, and initial point at the origin. Figure 1.5 shows several vectors, drawn as arrows beginning at the origin. In print, vectors are usually denoted by boldface letters such as \({\bf v}\) or \({\bf a}\). By hand, we usually write them as \(\vec{v}\) or simply as \(v\), possibly with a line or wavy line under it.

Using this definition of a vector, we associate with each vector \({\bf a}\) the point \((a_1,a_2,a_3)\) where \({\bf a}\) terminates, and conversely, we can associate a vector \({\bf v}\) with each point \((a_1,a_2,a_3)\) in space. Thus, we shall identify \({\bf a}\) with \((a_1,a_2,a_3)\) and write \({\bf a}=(a_1,a_2,a_3)\). For this reason, the elements of \({\mathbb R}^3\) not only are ordered triples of real numbers, but are also regarded as vectors. The triple (0, 0, 0) is denoted \({\bf 0}\). We call \(a_1,a_2\), and \(a_3\) the components of \({\bf a}\), or when we think of \({\bf a}\) as a point, its coordinates.

Two vectors \({\bf v}=(a_1,a_2,a_3)\) and \({\bf w}=(b_1,b_2,b_3)\) are equal if and only if \(a_1=b_1, a_2=b_2\), and \(a_3=b_3\). Geometrically this means that \({\bf v}\) and \({\bf w}\) have the same direction and the same length (or “magnitude”).

Geometrically, we define vector addition as follows. In the plane containing the vectors \({\bf v}=(a_1,a_2,a_3)\) and \({\bf w}=(b_1,b_2,b_3)\) (see Figure 1.6), form the parallelogram having \({\bf v}\) as one side and \({\bf w}\) as its adjacent side. The sum \({\bf v}+{\bf w}\) is the directed line segment along the diagonal of the parallelogram.

This geometric view of vector addition is useful in many physical situations, as we shall see in the next section. For an easily visualized example, consider a bird or an airplane flying through the air with velocity \({\bf v}_1\), but in the presence of a wind with velocity \({\bf v}_2\). The resultant velocity, \({\bf v}_1+{\bf v}_2,\) is what one sees; see Figure 1.7.

To show that our geometric definition of addition is consistent with our algebraic definition, we demonstrate that if \({\bf a}=(a_1, a_2, a_3)\) and \({\bf b}=(b_1, b_2, b_3)\), then \({\bf a}+{\bf b}=(a_1+b_1, a_2+b_2, a_3+b_3)\). We shall prove this result in the plane and leave the proof in three-dimensional space to the reader. Thus, we wish to show that if \({\bf a}=(a_1,a_2)\) and \({\bf b}=(b_1,b_2)\), then \({\bf a}+{\bf b}=(a_1+b_1,a_2+b_2)\).

In Figure 1.8 let \({\bf a}=(a_1,a_2)\) be the vector ending at the point A, and let \({\bf b}=(b_1,b_2)\) be the vector ending at point B. By definition, the vector \({\bf a}+{\bf b}\) ends at the vertex C of parallelogram OBCA. To verify that \({\bf a}+{\bf b}=(a_1+b_1,a_2+b_2)\), it suffices to show that the coordinates of C are \((a_1+b_1,a_2+b_2)\). The sides of the triangles OAD and BCG are parallel, and the sides OA and BC have equal lengths, which we write as OA \(=\) BC. These triangles are congruent, so BG \(=\) OD; since BGFE is a rectangle, EF \(=\) BG. Furthermore, OD \(=a_1\) and OE \(=b_1\). Hence, EF \(=\) BG \(=\) OD \(=a_1\). Since OF \(=\) EF \(+\) OE, it follows that OF \(=a_1+b_1\). This shows that the \(x\) coordinate of \({\bf a}+{\bf b}\) is \(a_1+b_1\). The proof that the \(y\) coordinate is \(a_2+b_2\) is analogous. This argument assumes A and B to be in the first quadrant, but similar arguments hold for the other quadrants.

Figure 1.9 illustrates another way of looking at vector addition: in terms of triangles rather than parallelograms. That is, we translate (without rotation) the directed line segment representing the vector \({\bf b}\) so that it begins at the end of the vector \({\bf b}\). The endpoint of the resulting directed segment is the endpoint of the vector \({\bf a}+{\bf b}\). We note that when \({\bf a}\) and \({\bf b}\) are collinear, the triangle collapses to a line segment, as in Figure 1.9.

In Figure 1.9 we have placed \({\bf a}\) and \({\bf b}\) head to tail. That is, the tail of \({\bf b}\) is placed at the head of \({\bf a}\), and the vector \({\bf a}+{\bf b}\) goes from the tail of \({\bf a}\) to the head of \({\bf b}\). If we do it in the other order, \({\bf b}+{\bf a}\), we get the same vector by going around the parallelogram the other way. Consistent with this figure, it is useful to let vectors “glide” or “slide,” keeping the same magnitude and direction. We want, in fact, to regard two vectors as the same if they have the same magnitude and direction. When we insist on vectors beginning at the origin, we will say that we have bound vectors. If we allow vectors to begin at other points, we will speak of free vectors or just vectors.

Scalar multiplication of vectors also has a geometric interpretation. If \(\alpha\) is a scalar and \({\bf a}\) a vector, we define \(\alpha{\bf a}\) to be the vector that is \(|\alpha|\) times as long as \({\bf a}\), with the same direction as \({\bf a}\) if \(\alpha>0\), but with the opposite direction if \(\alpha<0\). Figure 1.10 illustrates several examples.

Using an argument based on similar triangles, we find that if \({\bf a}=(a_1,a_2,a_3)\), and \(\alpha\) is a scalar, then \[ \alpha \ {\bf a}=(\alpha a_1,\alpha a_2,\alpha a_3). \]

That is, the geometric definition coincides with the algebraic one.

Given two vectors a and b, how do we represent the vector \({\bf b}-{\bf a}\) geometrically, that is, what is the geometry of vector subtraction? Because \({\bf a}+({\bf b}-{\bf a})={\bf b}\), we see that \({\bf b}-{\bf a}\) is the vector that we add to a to get b. In view of this, we may conclude that \({\bf b}-{\bf a}\) is the vector parallel to, and with the same magnitude as, the directed line segment beginning at the endpoint of a and terminating at the endpoint of b when a and b begin at the same point (see Figure 1.11).

The Standard Basis Vectors

To describe vectors in space, it is convenient to introduce three special vectors along the \(x,y\), and \(z\) axes: \[ \begin{array}{c} {\bf i}{:}\, \hbox{ the vector with components (1, 0, 0)}\\ {\bf j}{:}\, \hbox{ the vector with components (0, 1, 0)}\\ {\bf k}{:}\, \hbox{ the vector with components (0, 0, 1)}. \end{array} \]

These standard basis vectors are illustrated in Figure 1.15. In the plane we have the standard basis \({\bf i}\) and \({\bf j}\) with components (1, 0) and (0, 1).

Let \({\bf a}\) be any vector, and let \((a_1,a_2,a_3)\) be its components. Then \[ {\bf a}=a_1{\bf i}+a_2{\bf j}+a_3{\bf k}, \] because the right-hand side is given in components by \begin{eqnarray*} a_1(1,0,0)+a_2(0,1,0)+a_3(0,0,1) &=& (a_1,0,0)+(0,a_2,0)+(0,0,a_3)\\ &=& (a_1,a_2,a_3). \end{eqnarray*}

Thus, we can express every vector as a sum of scalar multiples of \({\bf i},{\bf j}\), and \({\bf k}\).

Addition and scalar multiplication may be written in terms of the standard basis vectors as follows: \[ (a_1{\bf i}+a_2{\bf j}+a_3{\bf k})+(b_1{\bf i}+b_2{\bf j}+b_3{\bf k}) = (a_1+b_1){\bf i}+(a_2+b_2){\bf j}+(a_3+b_3){\bf k} \] and \[ \alpha(a_1{\bf i}+a_2{\bf j}+a_3{\bf k}) = (\alpha a_1){\bf i}+(\alpha a_2){\bf j}+\alpha(a_3){\bf k}. \]

The Vector Joining Two Points

To apply vectors to geometric problems, it is useful to assign a vector to a pair of points in the plane or in space, as follows. Given two points P and Q, we can draw the vector \({\bf v}\) with tail P and head Q, as in Figure 1.17, where we write \(\overrightarrow{{\rm PQ}}\) for \(\bf v\) .

If P \(=(x,y,z)\) and Q\(=(x',y',z')\), then the vectors from the origin to P and Q are \({\bf a}=\) \(x{\bf i}\) \(+\) \(y{\bf j}\) \(+\) \(z{\bf k}\) and \({\bf b}=x'{\bf i}+y'{\bf j}+z'{\bf k}\), respectively, so the vector \(\overrightarrow{{\rm PQ}}\) is the difference \({\bf b}-{\bf a}=(x'-x){\bf i}+(\kern1pty'-y){\bf j}+(z'-z){\bf k}\). (See Figure 1.19.).

Geometry Theorems by Vector Methods

Many of the theorems of plane geometry can be proved by vector methods. Here is one example.

Equations of Lines

Planes and lines are geometric objects that can be represented by equations. We shall defer until Section 1.3 a study of equations representing planes. However, using the geometric interpretation of vector addition and scalar multiplication, we will now find the equation of a line l that passes through the endpoint of the vector \({\bf a}\), with the direction of a vector \({\bf v}\) (see Figure 1.22); that is, line \(l\) is parallel to the vector v.

As \(t\) varies through all real values, the points of the form \(t{\bf v}\) are all scalar multiples of the vector v, and therefore exhaust the points of the line passing through the origin in the direction of v. Because every point on \(l\) is the endpoint of the diagonal of a parallelogram with sides a and \(t{\bf v}\) for some real value of \(t\), we see that all the points on \(l\) are of the form \({\bf a}+t{\bf v}\). Thus, the line \(l\) may be expressed by the equation \({\bf l}(t)={\bf a}+t{\bf v}\). We say that \(l\) is expressed parametrically, with \(t\) the parameter. At \(t=0,{\bf l}(t)={\bf a}\). As \(t\) increases, the point \({\bf l}(t)\) moves away from a in the direction of v. As \(t\) decreases from \(t=0\) through negative values, \({\bf l}(t)\) moves away from a in the direction of \(-{\bf v}\).

There can be many equations of the same line. Some may be obtained by choosing, instead of \({\bf a}\), a different point on the given line, and forming the parametric equation of the line beginning at that point and in the direction of \({\bf v}\). For example, the endpoint of \({\bf a}+{\bf v}\) is on the line \({\bf l}(t)={\bf a}+t{\bf v}\), and thus, \({\bf l}_1(t)=({\bf a}+{\bf v})+t{\bf v}\) represents the same line. Still other equations may be obtained by observing that if \(\alpha \neq 0\), the vector \(\alpha {\bf v}\) has the same (or opposite) direction as \({\bf v}\). Thus, \({\bf l}_2(t)={\bf a}+t\alpha{\bf v}\) is another equation of \({\bf l}(t)={\bf a}+t{\bf v}\).

For example, both \({\bf l}(t)=(1,0,0)+(t,t,0)\) and \({\bf l}_1(s)=(0,-1,0)+(s,s,0)\) represent the same line since both are in the direction \({\bf i}+{\bf j}\) and both pass through the point (1, 0, 0); \({\bf l}\) passes through (1, 0, 0) at \(t=0\) and \({\bf l}_1\) passes through (1, 0, 0) at \(s=1\).

Therefore, the equation of a line is not uniquely determined. Nevertheless, it is customary to use the term “the equation of a line.” Keeping this in mind, let us derive the equation of a line passing through the endpoints of two given vectors \({\bf a}\) and \({\bf b}\). Because the vector \({\bf b}-{\bf a}\) is parallel to the directed line segment from \({\bf a}\) to \({\bf b}\), we calculate the parametric equation of the line passing through \({\bf a}\) in the direction of \({\bf b}-{\bf a}\) (Figure 1.24). Thus, \[ {\bf l}(t)={\bf a}+t({\bf b}-{\bf a});\qquad\hbox{that is},\qquad{\bf l}(t)=(1-t){\bf a}+t{\bf b}. \]

As \(t\) increases from 0 to 1, \(t({\bf b}-{\bf a})\) starts as the zero vector and increases in length (remaining in the direction of \({\bf b}-{\bf a}\)) until at \(t=1\) it is the vector \({\bf b}-{\bf a}\). Thus, for \({\bf l}(t)={\bf a}+t({\bf b}-{\bf a})\), as \(t\) increases from 0 to 1, the vector \({\bf l}(t)\) moves from the endpoint of \({\bf a}\) to the endpoint of \({\bf b}\) along the directed line segment from \({\bf a}\)to \({\bf b}\).

If P \(=(x_1,y_1,z_1)\) is the tip of \({\bf a}\) and Q \(=(x_2,y_2,z_2)\) is the tip of \({\bf b}\), then \({\bf v} = (x_2-x_1){\bf i} + (\kern1pty_2-y_1){\bf j} + (z_2-z_1){\bf k}\), and so the equations of the line are \begin{eqnarray*} x &=& x_1 +(x_2-x_1)t,\\ y &=& y_1+(\kern1pty_2-y_1)t,\\ z &=& z_1+(z_2-z_1)t.\\[-21pt] \end{eqnarray*}

By eliminating \(t\), these can be written as \[ \frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}. \]

The description of a line segment requires that the domain of the parameter \(t\) be restricted, as in the following example.

We can also give parametric descriptions of geometric objects other than lines.

As two different lines through the origin determine a plane through the origin, so do two nonparallel vectors. If we apply the same reasoning as in Example 17, we see that the entire plane formed by two nonparallel vectors \({\bf v}\) and \({\bf w}\) consists of all points of the form \(s{\bf v}+t{\bf w}\), where \(s\) and \(t\) can be any real numbers, as in Figure 1.27.

We have thus described the points in the plane by two parameters. For this reason, we say the plane is two-dimensional. Similarly, a line is called one-dimensional whether it lies in the plane or in space or is the real number line itself.

The plane determined by \({\bf v}\) and \({\bf w}\) is called the plane spanned by \({\bf v}\) and \({\bf w}\). When \({\bf v}\) is a scalar multiple of \({\bf w}\) and \({\bf w}\neq {\bf 0}\), then v and \({\bf w}\) are parallel and the plane degenerates to a straight line. When \({\bf v}={\bf w}={\bf 0}\) (i.e., both are zero vectors), we obtain a single point.

There are three particular planes that arise naturally in a coordinate system and that will be useful to us later. We call the plane spanned by vectors \({\bf i}\) and \({\bf j}\) the \(xy\) plane, the plane spanned by \({\bf j}\) and \({\bf k}\) the \(yz\) plane, and the plane spanned by \({\bf i}\) and \({\bf k}\) the \(xz\) plane. These planes are illustrated in Figure 1.28.