9.2: Vectors

 
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Representations of Vectors

Preview Activity $9.2.1$ shows how we can record the magnitude and direction of a change in position using an ordered pair of numbers $\langle x,y\rangle\text{.}$ There are many other quantities, such as force and velocity, that possess the attributes of magnitude and direction, and we will call each such quantity a vector.

Definition $9.2.1$

A vector is a quantity that possesses the attributes of magnitude and direction.

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 Figure $9.2.1$.

According to the definition, a vector possesses the attributes of length (magnitude) and direction; the vector's position, however, is not mentioned. Consequently, we regard as equal any two vectors having the same magnitude and direction, as shown at right in Figure $9.2.2$. 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. For instance, suppose that we would like to draw the vector $\langle 3, 4\rangle\text{,}$ which represents a horizontal change of three units and a vertical change of 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 $(3,4)\text{,}$ as illustrated at left in Figure $9.2.3$. A vector with its tail at the origin is said to be in standard position.

 

Alternatively, we may place the tail of the vector $\langle 3,4\rangle$ at another point, such as $Q(1,1)\text{.}$ After a displacement of three units to the right and four units up, the tip of the vector is at the point $R(4,5)$ (see the vector at right in Figure $9.2.3$).

In this example, the vector led to the directed line segment from $Q$ to $R\text{,}$ which we denote as $\overrightarrow{QR}\text{.}$ We may also turn the situation around: given the two points $Q$ and $R\text{,}$ we obtain the vector $\langle 3,4\rangle$ because we move horizontally three units and vertically four units to get from $Q$ to $R\text{.}$ In other words, $\overrightarrow{QR} = \langle 3,4\rangle\text{.}$ In general, the vector $\overrightarrow{QR}$ from the point $Q = (q_1, q_2)$ to $R = (r_1, r_2)$ is found by taking the difference of coordinates, so that

$$\overrightarrow{QR} = \langle r_1-q_1, r_2-q_2 \rangle.$$

We will use boldface letters to represent vectors, such as $v = \langle 3, 4 \rangle\text{,}$ to distinguish them from scalars. The entries of a vector are called its components; in the vector $\langle 3, 4 \rangle\text{,}$ the $x$ component is 3 and the $y$ component is 4. We use pointed brackets $\langle \ , \rangle$ and the term components to distinguish a vector from a point $( \ , )$ and its coordinates. There is, however, a close connection between vectors and points. Given a point $P\text{,}$ we will frequently consider the vector $\overrightarrow{OP}$ from the origin $O$ to $P\text{.}$ For instance, if $P=(3,4)\text{,}$ then $\overrightarrow{OP}=\langle 3,4\rangle$ as in Figure $9.2.4$. In this way, we think of a point $P$ as defining a vector $\overrightarrow{OP}$ whose components agree with the coordinates of $P\text{.}$ The vector $\overrightarrow{OP}$ is called the position vector of $P\text{.}$

While we often illustrate vectors in the plane since it is easier to draw pictures, different situations call for the use of vectors in three or more dimensions. For instance, a vector $\mathbf{v}$ in $n$-dimensional space, $\mathbb{R}^n\text{,}$ has $n$ components and may be represented as

The next activity will help us to become accustomed to vectors and operations on vectors in three dimensions.

Activity $9.2.2$

An article by C.Kenneth Tanner of the University of Georgia argues that, due to the concept of social distance, a secondary school classroom for 20 students should have 1344 square feet of floor space. Suppose a classroom is 32 feet by 42 feet by 8 feet. Set the origin $O$ of the classroom to be its center. In this classroom, a student is sitting on a chair whose seat is at location $A = (9, -6, -1.5)\text{,}$ an overhead projector is located at position $B = (0,1,3)\text{,}$ and the teacher is standing at point $C = (-2, 20, -4)\text{,}$ all distances measured in feet. Determine the components of the indicated vectors and explain in context what each represents.

a. $\overrightarrow{OA}$

b. $\overrightarrow{OB}$

c. $\overrightarrow{OC}$

d. $\overrightarrow{AB}$

e. $\overrightarrow{AC}$

f. $\overrightarrow{BC}$

Equality of Vectors

Because location is 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 $\mathbf{u}$ and $\mathbf{v} \text{,}$ we will want to be able to determine algebraically when $\mathbf{u}$ and $\mathbf{v}$ are equal. There is an obvious set of conditions that we use.

Equality of Vectors

Two vectors $\mathbf{u} = \langle u_1, u_2 \rangle$ and $\mathbf{v} = \langle v_1, v_2 \rangle$ in $\mathbb{R}^2$ are equal if and only if their corresponding components are equal: $u_1 = v_1$ and $u_2 = v_2\text{.}$ More generally, two vectors $\mathbf{u} = \langle u_1, u_2, \ldots, u_n\rangle$ and $\mathbf{v} = \langle v_1, v_2, \ldots, v_n \rangle$ in $\mathbb{R}^n$ are equal if and only if $u_i = v_i$ for each possible value of $i\text{.}$

Operations on Vectors

Vectors are not numbers, but we can now represent them 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 add two vectors together, as well as to multiply a vector by a scalar.

Activity $9.2.3$

Let $\mathbf{u} = \langle 2, 3 \rangle\text{,}$ $\mathbf{v} = \langle -1, 4 \rangle\text{.}$

1. Using the two specific vectors above, what is the natural way to define the vector sum $\mathbf{u} + \mathbf{v} \text{?}$
2. In general, how do you think the vector sum $\mathbf{a} + \mathbf{b}$ of vectors $\mathbf{a} = \langle a_1, a_2 \rangle$ and $\mathbf{b} = \langle b_1, b_2 \rangle$ in $\mathbb{R}^2$ should be defined? Write a formal definition of a vector sum based on your intuition.
3. In general, how do you think the vector sum $\mathbf{a} + \mathbf{b}$ of vectors $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$ and $\mathbf{b} = \langle b_1, b_2, b_3 \rangle$ in $\mathbb{R}^3$ should be defined? Write a formal definition of a vector sum based on your intuition.
4. Returning to the specific vector $\mathbf{v} = \langle -1, 4 \rangle$ given above, what is the natural way to define the scalar multiple $\frac{1}{2} \mathbf{v}\text{?}$
5. In general, how do you think a scalar multiple of a vector $\mathbf{a} = \langle a_1, a_2 \rangle$ in $\mathbb{R}^2$ by a scalar $c$ should be defined? how about for a scalar multiple of a vector $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$ in $\mathbb{R}^3$ by a scalar $c\text{?}$ Write a formal definition of a scalar multiple of a vector based on your intuition.

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, $v - u\text{,}$ as the sum of $v$ and $-1$ times $u\text{,}$ so that

Using vector addition and scalar multiplication, we will often represent vectors in terms of the special vectors $\mathbf{i} = \langle 1, 0 \rangle$ and $\mathbf{j} = \langle 0,1 \rangle\text{.}$ For instance, we can write the vector $\langle a, b \rangle$ in $\mathbb{R}^2$ as

which means that

In the context of $\mathbb{R}^3\text{,}$ we let $\mathbf{i} = \langle 1, 0, 0 \rangle\text{,}$ $\mathbf{j} = \langle 0,1,0 \rangle\text{,}$ and $\mathbf{k} = \langle 0,0,1 \rangle\text{,}$ and we can write the vector $\langle a, b, c \rangle$ in $\mathbb{R}^3$ as

The vectors $\mathbf{i} \text{,}$ $\mathbf{j} \text{,}$ and $\mathbf{k}$ are called the standard unit vectors (as we will learn momentarily, unit vectors have length 1), and are important in the physical sciences.

Properties of Vector Operations

We know that the scalar sum $1+2$ is equal to the scalar sum $2+1\text{.}$ 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 $\mathbf{v}$ and $\mathbf{u}$ and add them together and see what happens. Let $\mathbf{v} = \langle v_1, v_2 \rangle$ and $\mathbf{u} = \langle u_1, u_2 \rangle\text{.}$ Now we use the fact that $v_1\text{,}$ $v_2\text{,}$ $u_1\text{,}$ and $u_2$ are scalars, and that the addition of scalars is commutative to see that

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

Properties of vector operations

Let $\mathbf{v} \text{,}$ $\mathbf{u}\text{,}$ and $\mathbf{w}$ be vectors in $\mathbb{R}^n$ and let $a$ and $b$ be scalars. Then

1. $\displaystyle \mathbf{v} + \mathbf{u} = \mathbf{u} + \mathbf{v}$
2. $\displaystyle (\mathbf{v} + \mathbf{u}) + \mathbf{w} = \mathbf{v} + (\mathbf{u} + \mathbf{w})$
3. The vector $\mathbf{0} = \langle 0, 0, \ldots, 0 \rangle$ has the property that $\mathbf{v} + \mathbf{0} = \mathbf{v} \text{.}$ The vector $\mathbf{0}$ is called the zero vector.
4. $(-1)\mathbf{v} + \mathbf{v} = \mathbf{0} \text{.}$ The vector $(-1) \mathbf{v} = -\mathbf{v}$ is called the additive inverse of the vector $\mathbf{v} \text{.}$
5. $\displaystyle (a+b) \mathbf{v} = a \mathbf{v} + b \mathbf{v}$
6. $\displaystyle a(\mathbf{v} + \mathbf{u}) = a \mathbf{v} + a \mathbf{u}$
7. $\displaystyle (ab) \mathbf{v} = a(b \mathbf{v})$
8. $1 \mathbf{v} = \mathbf{v} \text{.}$

We verified the first property for vectors in $\mathbb{R}^2\text{;}$ it is straightforward to verify that the rest of the eight properties just noted hold for all vectors in $\mathbb{R}^n\text{.}$

Geometric Interpretation of Vector Operations

Next, we explore a geometric interpretation of vector addition and scalar multiplication that allows us to visualize these operations. Let $\mathbf{u} = \langle 4, 6 \rangle$ and $\mathbf{v} = \langle 3, -2 \rangle\text{.}$ Then $\mathbf{w} = \mathbf{u} + \mathbf{v} = \langle 7, 4 \rangle\text{,}$ as shown on the left in Figure $9.2.5$.

If we think of these vectors as displacements in the plane, we find a geometric way to envision vector addition. For instance, the vector $\mathbf{u} + \mathbf{v}$ will represent the displacement obtained by following the displacement $\mathbf{u}$ with the displacement $\mathbf{v} \text{.}$ We may picture this by placing the tail of $\mathbf{v}$ at the tip of $\mathbf{u} \text{,}$ as seen in the center of Figure $9.2.5$.

Of course, vector addition is commutative so we obtain the same sum if we place the tail of $\mathbf{u}$ at the tip of $\mathbf{v} \text{.}$ We therefore see that $\mathbf{u}+\mathbf{v}$ appears as the diagonal of the parallelogram determined by $\mathbf{u}$ and $\mathbf{v}\text{,}$ as shown at right in Figure $9.2.5$.

Vector subtraction has a similar interpretation. At left in Figure $9.2.6$ we see vectors $\mathbf{u}\text{,}$ $\mathbf{v}\text{,}$ and $\mathbf{w} =\mathbf{u} + \mathbf{v} \text{.}$ If we rewrite $\mathbf{v} = \mathbf{w} - \mathbf{u} \text{,}$ we have the arrangement shown at right in Figure $9.2.6$. In other words, to form the difference $\mathbf{w} -\mathbf{u} \text{,}$ we draw a vector from the tip of $\mathbf{u}$ to the tip of $\mathbf{w} \text{.}$

 

In a similar way, we may geometrically represent a scalar multiple of a vector. For instance, if $\mathbf{v}=\langle 2,3\rangle\text{,}$ then $2\mathbf{v} = \langle 4,6\rangle\text{.}$ As shown in Figure $9.2.7$, multiplying $\mathbf{v}$ by 2 leaves the direction unchanged, but stretches $\mathbf{v}$ by 2. Also, $-2\mathbf{v} = \langle -4, -6\rangle\text{,}$ which shows that multiplying by a negative scalar gives a vector pointing in the opposite direction of $\mathbf{v} \text{.}$

Activity $9.2.4$

 

Suppose that $\mathbf{u}$ and $\mathbf{v}$ are the vectors shown in Figure $9.2.8$.

1. On the axes at left in Figure $9.2.8$, sketch the vectors $\mathbf{u} + \mathbf{v} \text{,}$ $\mathbf{v} - \mathbf{u} \text{,}$ $2\mathbf{u} \text{,}$ $-2\mathbf{u}\text{,}$ and $-3\mathbf{v}\text{.}$
2. What is $0\mathbf{v} \text{?}$
3. On the axes at right in Figure $9.2.8$, sketch the vectors $-3\mathbf{v} \text{,}$ $-2\mathbf{v} \text{,}$ $-1\mathbf{v} \text{,}$ $2\mathbf{v} \text{,}$ and $3\mathbf{v} \text{.}$
4. Give a geometric description of the set of terminal points of the vectors $t\mathbf{v}$ where $t$ is any scalar.
5. On the set of axes at right in Figure $9.2.8$, sketch the vectors $\mathbf{u} - 3\mathbf{v} \text{,}$ $\mathbf{u} - 2\mathbf{v}\text{,}$ $\mathbf{u} -\mathbf{v} \text{,}$ $\mathbf{u} + \mathbf{v} \text{,}$ and $\mathbf{u} + 2\mathbf{v} \text{.}$
6. Give a geometric description of the set of terminal points of the vectors $\mathbf{u} + t\mathbf{v}$ where $t$ is any scalar.

The Magnitude of a Vector

By definition, vectors have both direction and magnitude (or length). We now investigate how to calculate the magnitude of a vector. Since a vector $\mathbf{v}$ 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 $\mathbf{v}$ and is denoted $|\mathbf{v}|\text{.}$

Activity $9.2.5$

 

1. Let $A = (2,3)$ and $B = (4,7)\text{,}$ as shown at left in Figure $9.2.9$. Compute $|\overrightarrow{AB}|\text{.}$
2. Let $\mathbf{v} = \langle v_1, v_2 \rangle$ be the vector in $\mathbb{R}^2$ with components $v_1$ and $v_2$ as shown at right in Figure $9.2.9$. Use the distance formula to find a general formula for $|\mathbf{v}|\text{.}$
3. Let $\mathbf{v} = \langle v_1, v_2, v_3 \rangle$ be a vector in $\mathbb{R}^3\text{.}$ Use the distance formula to find a general formula for $|\mathbf{v}|\text{.}$
4. Suppose that $\mathbf{u} = \langle 2,3\rangle$ and $\mathbf{v} = \langle -1,2\rangle\text{.}$ Find $|\mathbf{u}|\text{,}$ $|\mathbf{v}|\text{,}$ and $|\mathbf{u}+\mathbf{v}|\text{.}$ Is it true that $|\mathbf{u} + \mathbf{v}| = |\mathbf{u}|+|\mathbf{v}|\text{?}$
5. Under what conditions will $|\mathbf{u}+\mathbf{v}| = |\mathbf{u}|+|\mathbf{v}|\text{?}$ (Hint: Think about how $\mathbf{u} \text{,}$ $\mathbf{v} \text{,}$ and $\mathbf{u}+\mathbf{v}$ form the sides of a triangle.)
6. With the vector $\mathbf{u} = \langle 2,3\rangle\text{,}$ find the lengths of $2\mathbf{u}\text{,}$ $3\mathbf{u}\text{,}$ and $-2\mathbf{u}\text{,}$ respectively, and use proper notation to label your results.
7. If $t$ is any scalar, how is $|t\mathbf{u}|$ related to $|\mathbf{u}|\text{?}$
8. A unit vector is a vector whose magnitude is 1. Of the vectors $\textbf{i}\text{,}$ $\textbf{j}\text{,}$ and $\textbf{i} + \textbf{ j}\text{,}$ which are unit vectors?
9. Find a unit vector $\mathbf{v}$ whose direction is the same as $\mathbf{u} = \langle 2, 3\rangle\text{.}$ (Hint: Consider the result of part (g).)

Summary

• A vector is an object that possesses the attributes of magnitude and direction. Examples of vector quantities are position, velocity, acceleration, and force.
• Two vectors are equal if they have the same direction and magnitude. Notice that position is not considered, so a vector is independent of its location.
• If $\mathbf{u} = \langle u_1, u_2, \ldots, u_n \rangle$ and $\mathbf{v} = \langle v_1, v_2, \ldots, v_n \rangle$ are two vectors in $\mathbb{R}^n\text{,}$ then their vector sum is the vector $$\mathbf{u} + \mathbf{v} = \langle u_1+v_1, u_2+v_2, \ldots, u_n+v_n \rangle.$$
• If $\mathbf{u} = \langle u_1, u_2, \ldots, u_n \rangle$ is a vector in $\mathbb{R}^n$ and $c$ is a scalar, then the scalar multiple $c\mathbf{u}$ is the vector $$c\mathbf{u} = \langle cu_1, cu_2, \ldots, cu_n \rangle.$$
• The magnitude of the vector $\mathbf{v} = \langle v_1, v_2, \ldots, v_n \rangle$ in $\mathbb{R}^n$ is the scalar $$|\mathbf{v}| = \sqrt{v_1^2+v_2^2+ \cdots + v_n^2}.$$
• A vector $\mathbf{u}$ is a unit vector provided that $|\mathbf{u}| = 1\text{.}$ If $\mathbf{v}$ is a nonzero vector, then the vector $\frac{v}{|v|}$ is a unit vector with the same direction as $\mathbf{v} \text{.}$