Learning Outcomes

A.

Find the position vector of a point in ${ℝ}^n$.

The notation ${ℝ}^n$ refers to the collection of ordered lists of $n$ real numbers, that is

In this chapter, we take a closer look at vectors in ${ℝ}^n$. First, we will consider what ${ℝ}^n$ looks like in more detail. Recall that the point given by $0=\left(0,\cdots ,0\right)$ is called the origin.

Now, consider the case of ${ℝ}^n$ for $n=1.$ Then from the definition we can identify $ℝ$ with points in ${ℝ}^1$ as follows:

Hence, $ℝ$ is defined as the set of all real numbers and geometrically, we can describe this as all the points on a line.

Now suppose $n=2$. Then, from the definition,

Consider the familiar coordinate plane, with an $x$ axis and a $y$ axis. Any point within this coordinate plane is identified by where it is located along the $x$ axis, and also where it is located along the $y$ axis. Consider as an example the following diagram.

Hence, every element in ${ℝ}^2$ is identified by two components, $x$ and $y$, in the usual manner. The coordinates $x,y$ (or $x_1$,$x_2$) uniquely determine a point in the plan. Note that while the definition uses $x_1$ and $x_2$ to label the coordinates and you may be used to $x$ and $y$, these notations are equivalent.

Now suppose $n=3$. You may have previously encountered the $3$-dimensional coordinate system, given by

Points in ${ℝ}^3$ will be determined by three coordinates, often written $\left(x,y,z\right)$ which correspond to the $x$, $y$, and $z$ axes. We can think as above that the first two coordinates determine a point in a plane. The third component determines the height above or below the plane, depending on whether this number is positive or negative, and all together this determines a point in space. You see that the ordered triples correspond to points in space just as the ordered pairs correspond to points in a plane and single real numbers correspond to points on a line.

The idea behind the more general ${ℝ}^n$ is that we can extend these ideas beyond $n=3.$ This discussion regarding points in ${ℝ}^n$ leads into a study of vectors in ${ℝ}^n$. While we consider ${ℝ}^n$ for all $n$, we will largely focus on $n=2,3$ in this section.

Definition D.1: The Position Vector

Let $P=\left(p_1,\cdots ,p_n\right)$ be the coordinates of a point in ${ℝ}^n.$ Then the vector $\overrightarrow{0P}$ with its tail at $0=\left(0,\cdots ,0\right)$ and its tip at $P$ is called the position vector of the point $P$. We write

For this reason we may write both $P=\left(p_1,\cdots ,p_n\right)\in {ℝ}^n$ and $\overrightarrow{0P}={\left[p_1\cdots p_n\right]}^T\in {ℝ}^n$.

This definition is illustrated in the following picture for the special case of ${ℝ}^3$.

Thus every point $P$ in ${ℝ}^n$ determines its position vector $\overrightarrow{0P}$. Conversely, every such position vector $\overrightarrow{0P}$ which has its tail at $0$ and point at $P$ determines the point $P$ of ${ℝ}^n$.

Now suppose we are given two points, $P,Q$ whose coordinates are $\left(p_1,\cdots ,p_n\right)$ and $\left(q_1,\cdots ,q_n\right)$ respectively. We can also determine the position vector from $P$ to $Q$ (also called the vector from $P$ to $Q$) defined as follows.

Now, imagine taking a vector in ${ℝ}^n$ and moving it around, always keeping it pointing in the same direction as shown in the following picture.

After moving it around, it is regarded as the same vector. Each vector, $\overrightarrow{0P}$ and $\overrightarrow{AB}$ has the same length (or magnitude) and direction. Therefore, they are equal.

Definition D.2: Vectors in ${ℝ}^n$

Let ${ℝ}^n=\left\{\left(x_1,\cdots ,x_n\right):x_j\in ℝ\text{ for }j=1,\cdots ,n\right\}.$ Then,

is called a vector. Vectors have both size (magnitude) and direction. The numbers $x_j$ are called the components of $\overrightarrow{x}$.

Using this notation, we may use $\overrightarrow{p}$ to denote the position vector of point $P$. Notice that in this context, $\overrightarrow{p}=\overrightarrow{0P}$. These notations may be used interchangeably.

You can think of the components of a vector as directions for obtaining the vector. Consider $n=3$. Draw a vector with its tail at the point $\left(0,0,0\right)$ and its tip at the point $\left(a,b,c\right)$. This vector it is obtained by starting at $\left(0,0,0\right)$, moving parallel to the $x$ axis to $\left(a,0,0\right)$ and then from here, moving parallel to the $y$ axis to $\left(a,b,0\right)$ and finally parallel to the $z$ axis to $\left(a,b,c\right).$ Observe that the same vector would result if you began at the point $\left(d,e,f\right)$, moved parallel to the $x$ axis to $\left(d+a,e,f\right),$ then parallel to the $y$ axis to $\left(d+a,e+b,f\right),$ and finally parallel to the $z$ axis to $\left(d+a,e+b,f+c\right)$. Here, the vector would have its tail sitting at the point determined by $A=\left(d,e,f\right)$ and its point at $B=\left(d+a,e+b,f+c\right).$ It is the same vector because it will point in the same direction and have the same length. It is like you took an actual arrow, and moved it from one location to another keeping it pointing the same direction.

We conclude this section with a brief discussion regarding notation. In previous sections, we have written vectors as columns, or $n\times 1$ matrices. For convenience in this chapter we may write vectors as the transpose of row vectors, or $1\times n$ matrices. These are of course equivalent and we may move between both notations. Therefore, recognize that

Notice that two vectors $\overrightarrow{u}={\left[u_1\cdots u_n\right]}^T$ and $\overrightarrow{v}={\left[v_1\cdots v_n\right]}^T$ are equal if and only if all corresponding components are equal. Precisely,

Thus ${\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 4\end{array}\right]}^T\in {ℝ}^3$ and ${\left[\begin{array}{ccc}\hfill 2& \hfill 1& \hfill 4\end{array}\right]}^T\in {ℝ}^3$ but ${\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 4\end{array}\right]}^T\ne {\left[\begin{array}{ccc}\hfill 2& \hfill 1& \hfill 4\end{array}\right]}^T$ because, even though the same numbers are involved, the order of the numbers is different.

For the specific case of ${ℝ}^3$, there are three special vectors which we often use. They are given by

We can write any vector $\overrightarrow{u}={\left[\begin{array}{ccc}\hfill u_1& \hfill u_2& \hfill u_3\end{array}\right]}^T$ as a linear combination of these vectors, written as $\overrightarrow{u}=u_1\overrightarrow{i}+u_2\overrightarrow{j}+u_3\overrightarrow{k}$. This notation will be used throughout this chapter.

Learning Outcomes

A.

Understand vector addition and scalar multiplication, algebraically.

B.

Introduce the notion of linear combination of vectors.

Addition and scalar multiplication are two important algebraic operations done with vectors. Notice that these operations apply to vectors in ${ℝ}^n$, for any value of $n$. We will explore these operations in more detail in the following sections.

Definition D.3: Addition of Vectors in ${ℝ}^n$

If $\overrightarrow{u}=\left[\begin{array}{c}\hfill u_1\hfill \\ \hfill ⋮\hfill \\ \hfill u_n\hfill \end{array}\right],\overrightarrow{v}=\left[\begin{array}{c}\hfill v_1\hfill \\ \hfill ⋮\hfill \\ \hfill v_n\hfill \end{array}\right]\in {ℝ}^n$ then $\overrightarrow{u}+\overrightarrow{v}\in {ℝ}^n$ and is defined by

To add vectors, we simply add corresponding components. Therefore, in order to add vectors, they must be the same size.

Theorem D.4: Properties of Vector Addition

The following properties hold for vectors $\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}\in {ℝ}^n$.

• The Commutative Law of Addition

  $$\overrightarrow{u}+\overrightarrow{v}=\overrightarrow{v}+\overrightarrow{u}$$

• The Associative Law of Addition

  $$\left(\overrightarrow{u}+\overrightarrow{v}\right)+\overrightarrow{w}=\overrightarrow{u}+\left(\overrightarrow{v}+\overrightarrow{w}\right)$$

• The Existence of an Additive Identity

  $$\overrightarrow{u}+\overrightarrow{0}=\overrightarrow{u}$$

  (D.1)

• The Existence of an Additive Inverse

  $$\overrightarrow{u}+\left(-\overrightarrow{u}\right)=\overrightarrow{0}$$

Definition D.5: Scalar Multiplication of Vectors in ${ℝ}^n$

If $\overrightarrow{u}\in {ℝ}^n$ and $k\in ℝ$ is a scalar, then $k\overrightarrow{u}\in {ℝ}^n$ is defined by

Theorem D.6: Properties of Scalar Multiplication

The following properties hold for vectors $\overrightarrow{u},\overrightarrow{v}\in {ℝ}^n$ and $k,p$ scalars.

• The Distributive Law over Vector Addition

  $$k\left(\overrightarrow{u}+\overrightarrow{v}\right)=k\overrightarrow{u}+k\overrightarrow{v}$$

• The Distributive Law over Scalar Addition

  $$\left(k+p\right)\overrightarrow{u}=k\overrightarrow{u}+p\overrightarrow{u}$$

• The Associative Law for Scalar Multiplication

  $$k\left(p\overrightarrow{u}\right)=\left(kp\right)\overrightarrow{u}$$

• Rule for Multiplication by $1$

  $$1\overrightarrow{u}=\overrightarrow{u}$$

Definition D.7: Linear Combination

A vector $\overrightarrow{v}$ is said to be a linear combination of the vectors ${\overrightarrow{u}}_1,\cdots ,{\overrightarrow{u}}_n$ if there exist scalars, $a_1,\cdots ,a_n$ such that

For example,

Thus we can say that

is a linear combination of the vectors

Learning Outcomes

A.

Understand vector addition, geometrically.

Recall that an element of ${ℝ}^n$ is an ordered list of numbers. For the specific case of $n=2,3$ this can be used to determine a point in two or three dimensional space. This point is specified relative to some coordinate axes.

Consider the case $n=3$. Recall that taking a vector and moving it around without changing its length or direction does not change the vector. This is important in the geometric representation of vector addition.

Suppose we have two vectors, $\overrightarrow{u}$ and $\overrightarrow{v}$ in ${ℝ}^3$. Each of these can be drawn geometrically by placing the tail of each vector at $0$ and its point at $\left(u_1,u_2,u_3\right)$ and $\left(v_1,v_2,v_3\right)$ respectively. Suppose we slide the vector $\overrightarrow{v}$ so that its tail sits at the point of $\overrightarrow{u}$. We know that this does not change the vector $\overrightarrow{v}$. Now, draw a new vector from the tail of $\overrightarrow{u}$ to the point of $\overrightarrow{v}$. This vector is $\overrightarrow{u}+\overrightarrow{v}$.

Definition D.8: Geometry of Vector Addition

Let $\overrightarrow{u}$ and $\overrightarrow{v}$ be two vectors. Slide $\overrightarrow{v}$ so that the tail of $\overrightarrow{v}$ is on the point of $\overrightarrow{u}$. Then draw the arrow which goes from the tail of $\overrightarrow{u}$ to the point of $\overrightarrow{v}$. This arrow represents the vector $\overrightarrow{u}+\overrightarrow{v}$.

This definition is illustrated in the following picture in which $\overrightarrow{u}+\overrightarrow{v}$ is shown for the special case $n=3$.

Example D.9: Graphing Vector Addition

Consider the following picture of vectors $\overrightarrow{u}$ and $\overrightarrow{v}$.

Sketch a picture of $\overrightarrow{u}+\overrightarrow{v},\overrightarrow{u}-\overrightarrow{v}.$

Solution

We will first sketch $\overrightarrow{u}+\overrightarrow{v}.$ Begin by drawing $\overrightarrow{u}$ and then at the point of $\overrightarrow{u}$, place the tail of $\overrightarrow{v}$ as shown. Then $\overrightarrow{u}+\overrightarrow{v}$ is the vector which results from drawing a vector from the tail of $\overrightarrow{u}$ to the tip of $\overrightarrow{v}$.

Next consider $\overrightarrow{u}-\overrightarrow{v}.$ This means $\overrightarrow{u}+\left(-\overrightarrow{v}\right).$ From the above geometric description of vector addition, $-\overrightarrow{v}$ is the vector which has the same length but which points in the opposite direction to $\overrightarrow{v}$. Here is a picture.

Learning Outcomes

A.

Find the length of a vector and the distance between two points in ${ℝ}^n$.

B.

Find the corresponding unit vector to a vector in ${ℝ}^n$.

In this section, we explore what is meant by the length of a vector in ${ℝ}^n$. We develop this concept by first looking at the distance between two points in ${ℝ}^n$.

First, we will consider the concept of distance for $ℝ$, that is, for points in ${ℝ}^1$. Here, the distance between two points $P$ and $Q$ is given by the absolute value of their difference. We denote the distance between $P$ and $Q$ by $d(P,Q)$ which is defined as

Consider now the case for $n=2$, demonstrated by the following picture.

Now suppose $n=3$ and let $P=\left(p_1,p_2,p_3\right)$ and $Q=\left(q_1,q_2,q_3\right)$ be two points in ${ℝ}^3.$ Consider the following picture in which the solid line joins the two points and a dotted line joins the points $\left(q_1,q_2,q_3\right)$ and $\left(p_1,p_2,q_3\right).$

while the length of the line joining $\left(p_1,p_2,q_3\right)$ to $\left(p_1,p_2,p_3\right)$ is just $\left|p_3-q_3\right|.$ Therefore, by the Pythagorean Theorem again, the length of the line joining the points $P=\left(p_1,p_2,p_3\right)$ and $Q=\left(q_1,q_2,q_3\right)$ equals

Definition D.10: Distance Between Points

Let $P=\left(p_1,\cdots ,p_n\right)$ and $Q=\left(q_1,\cdots ,q_n\right)$ be two points in ${ℝ}^n$. Then the distance between these points is defined as

This is called the distance formula. We may also write $\left|P-Q\right|$ as the distance between $P$ and $Q$.

Example D.11: Distance Between Points

Find the distance between the points $P$ and $Q$ in ${ℝ}^4$, where $P$ and $Q$ are given by

and

Solution

Theorem D.12: Properties of Distance

Example D.13: The Plane Between Two Points

Describe the points in ${ℝ}^3$ which are at the same distance between $\left(1,2,3\right)$ and $\left(0,1,2\right).$

Solution

Definition D.14: Length of a Vector

Let $\overrightarrow{u}={\left[u_1\cdots u_n\right]}^T$ be a vector in ${ℝ}^n$. Then, the length of $\overrightarrow{u}$, written $\parallel \overrightarrow{u}\parallel$ is given by

Definition D.15: Unit Vector

Let $\overrightarrow{u}$ be a vector in ${ℝ}^n$. Then, we call $\overrightarrow{u}$ a unit vector if it has length 1, that is if

Let $\overrightarrow{v}$ be a vector in ${ℝ}^n$. Then, the vector $\overrightarrow{u}$ which has the same direction as $\overrightarrow{v}$ but length equal to $1$ is the corresponding unit vector of $\overrightarrow{v}$. This vector is given by

Example D.16: Finding a Unit Vector

Let $\overrightarrow{v}$ be given by

Find the unit vector $\overrightarrow{u}$ which has the same direction as $\overrightarrow{v}$ .

Solution

Learning Outcomes

A.

Understand scalar multiplication, geometrically.

Example D.17: Graphing Scalar Multiplication

Consider the vectors $\overrightarrow{u}$ and $\overrightarrow{v}$ drawn below.

Draw $-\overrightarrow{u}$, $2\overrightarrow{v}$, and $-\frac{1}{2}\overrightarrow{v}$.

Solution

In order to find $-\overrightarrow{u}$, we preserve the length of $\overrightarrow{u}$ and simply reverse the direction. For $2\overrightarrow{v}$, we double the length of $\overrightarrow{v}$, while preserving the direction. Finally $-\frac{1}{2}\overrightarrow{v}$ is found by taking half the length of $\overrightarrow{v}$ and reversing the direction. These vectors are shown in the following diagram.

Example D.18: Graphing a Linear Combination of Vectors

Consider the following picture of the vectors $\overrightarrow{u}$ and $\overrightarrow{v}$

Sketch a picture of $\overrightarrow{u}+2\overrightarrow{v},\overrightarrow{u}-\frac{1}{2}\overrightarrow{v}.$

Solution

The two vectors are shown below.

Learning Outcomes

A.

Compute the dot product of vectors, and use this to compute vector projections.

Definition D.19: Dot Product

Let $\overrightarrow{u},\overrightarrow{v}$ be two vectors in ${ℝ}^n$. Then we define the dot product $\overrightarrow{u}\bullet \overrightarrow{v}$ as

The dot product $\overrightarrow{u}\bullet \overrightarrow{v}$ is sometimes denoted as $(\overrightarrow{u},\overrightarrow{v})$ where a comma replaces $\bullet$. It can also be written as $⟨\overrightarrow{u},\overrightarrow{v}⟩$. If we write the vectors as column or row matrices, it is equal to the matrix product $\overrightarrow{v}{\overrightarrow{w}}^T$.

Example D.20: Compute a Dot Product

Find $\overrightarrow{u}\bullet \overrightarrow{v}$ for

Solution

Example D.21: Length of a Vector

Find the length of

That is, find $\parallel \overrightarrow{u}\parallel .$

Solution

Theorem D.22: Cauchy Schwarz Inequality

The dot product satisfies the inequality

Furthermore equality is obtained if and only if one of $\overrightarrow{u}$ or $\overrightarrow{v}$ is a scalar multiple of the other.

Theorem D.23: Triangle Inequality

For $\overrightarrow{u},\overrightarrow{v}\in {ℝ}^n$

and equality holds if and only if one of the vectors is a non-negative scalar multiple of the other.

Also

Given two vectors, $\overrightarrow{u}$ and $\overrightarrow{v}$, the included angle is the angle between these two vectors which is given by $𝜃$ such that $0\le 𝜃\le \pi$. The dot product can be used to determine the included angle between two vectors. Consider the following picture where $𝜃$ gives the included angle.

In words, the dot product of two vectors equals the product of the magnitude (or length) of the two vectors multiplied by the cosine of the included angle. Note this gives a geometric description of the dot product which does not depend explicitly on the coordinates of the vectors.

Example D.24: Find the Angle Between Two Vectors

Find the angle between the vectors given by

Solution

Example D.25: Find the Angle Between Two Lines

Find the angle between the two lines

and

Solution

You can verify that these lines do not intersect, but as discussed above this does not matter and we simply find the smallest angle between any directions vectors for these lines.

To do so we first find the angle between the direction vectors given above:

In order to find the angle, we solve the following equation for $𝜃$

to obtain $\cos 𝜃=-\frac{1}{2}$ and since we choose included angles between $0$ and $\pi$ we obtain $𝜃=\frac{2\pi }{3}$.

Now the angles between any two direction vectors for these lines will either be $\frac{2\pi }{3}$ or its complement $\varphi =\pi -\frac{2\pi }{3}=\frac{\pi }{3}$. We choose the smaller angle, and therefore conclude that the angle between the two lines is $\frac{\pi }{3}$.

Example D.26: Using Geometric Description to Find a Dot Product

Let $\overrightarrow{u},\overrightarrow{v}$ be vectors with $\parallel \overrightarrow{u}\parallel =3$ and $\parallel \overrightarrow{v}\parallel =4$. Suppose the angle between $\overrightarrow{u}$ and $\overrightarrow{v}$ is $\pi ∕3$. Find $\overrightarrow{u}\bullet \overrightarrow{v}$.

Solution

Example D.27: Determine if Two Vectors are Perpendicular

Determine whether the two vectors,

are perpendicular.

Solution

In order to determine if these two vectors are perpendicular, we compute the dot product. This is given by