Lecture Notes

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Week 1: Introduction to 2D and 3D co-ordinate spaces; introduction to vectors, vector spaces, vector operations

Sujeet Akula

Review, Notation

Functions are mappings. More specifically, for a given element of a function's domain, the function will map this element to an element of the function's co-domain. These aspects of the function are specified using the following notation: for a given function $f$, with domain $X$, and co-domain $Y$, we write \(f: X\to Y\). Remember that the domain serves as the set of "inputs'' to the function and the co-domain serves as the set of possible "outputs'' from the function. So far, you have studied single-variable functions of a single variable, which is technically written for a function $f$ as $f: \R \to \R$. This means that $f$ takes as an input, an element from the set $\R$, which is the set of all real numbers. And, the output will also be an element from the set $\R$. An example of this is the function $f(x) = x^3 + 2x$. For this example, let's consider $f(2.1)=13.461$: here, $f$ "maps'' the number $2.1$ (a real number) to $13.461$ (another real number). Geometrically, you can visualize the set $\R$ as the number line. In order to extend the number line into 2D and 3D spaces, the set product is employed. The set $\R \times \R$ which is written as $\Rtwo$, is the set of all pairs of elements in $\R$. This means that elements of $\Rtwo$ look like $(1.32, 7.5)$ and $(7.511, 3.2)$, etc. The set $\Rtwo$ is visualized geometrically as a plane. Similarly $\R \times \R \times \R$, also written as $\Rthree$, is the set of all triples of elements in $\R$, and $\Rthree$ is visualized as 3D space. In this course, we will consider the calculus of several-variable functions of several variables. This means, functions which are written as $f: \R^m\to\R^n$. (Here, $m$ and $n$ are positive integers.) If $m=1$ and $n=2$, i.e., $f:\R\to\Rthree$, then $f$ is a mapping from the number-line to 3D space. An example of such a function is the 3D vector representing the position $\vect{x}$ of a particle at a given time $t$.

2D and 3D Co-ordinate Spaces

Given that we will be studying functions that map several variables to several variables, it is necessary to introduce different co-ordinate systems, that may be better suited for analysis of higher dimensional geometries.

Two Dimensions

These co-ordinate systems can describe points that lie on a plane, a representation of $\Rtwo$.

2D Cartesian (or Rectangular) Co-ordinates

This is the familiar co-ordinate system, where a single point in the plane is marked and labeled as the "origin'', or simply $O$. Intersecting at this point are the two perpendicular axes for $x$ and $y$. Then, a given point $P$ on this plane can be labeled as $(x,y)$, where $x$ is the distance along the $x$ axis to $P$ and $y$ is the distance along the $y$ axis to $P$. Of course, by now, this system should be intimately familiar to you.

Polar Co-ordinates

Here, we again begin with the origin $O$, but polar co-ordinates only requires a single half-axis called the "polar axis'' emerging from the origin (i.e., a ray). This is taken to be the positive half of the $x$-axis. Then, a given point $P$ may be labeled by $(r,\theta)$, where $r$ is the distance from $O$ to $P$, and $\theta$ is the angle measured from the polar axis to $P$ (or more accurately, to the ray from the origin that also passes through $P$). Following the so-called right-handed convention, $\theta$ is positive for a counter-clockwise angle, and negative for a clockwise angle. We have the following equations to convert Cartesian to polar co-ordinates: \begin{equation} \begin{cases} r^2 = x^2 + y^2 \\ \tan\theta = \dfrac{y}{x} \end{cases} . \end{equation} Note that $r$ must always be non-negative since it is a distance, and that the equation for $\theta$ can be inverted as $\theta = \arctan\dfrac{y}{x}$, but the signs of $y$ and $x$ must be considered. For example, the point $(-1,-1)$ gives $\theta = \arctan \dfrac{-1}{-1} = \arctan 1 = \dfrac{\pi}{4}$, but it should be $\theta=\dfrac{5\pi}{4}$. (Actually, $\arctan 1 = \dfrac{\pi}{4} + \pi n, n = 0, 1, 2, \dots$ .) So, one must use the correct value of the arc-tangent in order to have the correct polar angle. (This should be something that you are already familiar with.) To convert from polar to Cartesian, we use: \begin{equation} \begin{cases} x = r\cos\theta \\ y = r\sin\theta \end{cases} . \end{equation}

Three Dimensions

These co-ordinate systems can describe points that lie in 3D space, a representation of $\Rthree$.

3D Cartesian (or Rectangular) Co-ordinates

Here, we begin with the 2D Cartesian system, with $x$ and $y$ axes, and create a third axis through the origin that is orthogonal to the $x$--$y$ plane, the $z$-axis. So a point $P$ will be labeled as $(x,y,z)$, where $x$ and $y$ are as before, and now $z$ is as you may suspect, the distance to $P$ from $O$ along the $z$-axis.

Cylindrical Co-ordinates

This is a trivial generalization of the polar co-ordinate system to three dimensions, where the $z$-axis is introduced orthogonal to the plane spanned by $r$ and $\theta$ so that a point $P$ is labeled by $(r,\theta,z)$. Again, $z$ is the distance to $P$ from $O$ along the $z$-axis. Note that $r$ is the distance from $O$ to $P$ that on the $x$--$y$ plane. To convert from (3D) Cartesian to cylindrical we have the following equations: \begin{equation} \begin{cases} r^2 = x^2 + y^2 \\ \tan\theta = \dfrac{y}{x} \\ z = z \end{cases} . \end{equation} To convert from cylindrical to (3D) Cartesian we have the following equations: \begin{equation} \begin{cases} x = r\cos\theta \\ y = r\sin\theta \\ z = z \end{cases} . \end{equation}

Spherical Co-ordinates

This is a non-trivial representation of 3D space, using one distance and two angles. Here a point $P$ is labeled by $(\rho,\theta,\phi)$, where $\rho$ is the distance from $O$ to $P$ in 3D space, $\theta$ is the angle from the positive $x$-axis to $P$, and $\phi$ is the angle from the $z$-axis to $P$. To convert from (3D) Cartesian to spherical we have the following equations: \begin{equation} \begin{cases} \rho^2 = x^2 + y^2 + z^2 \\ \tan\theta = \dfrac{y}{x} \\ \cos\phi = \dfrac{z}{\sqrt{x^2 + y^2 + z^2}} \end{cases} . \end{equation} To convert from spherical to (3D) Cartesian we have the following equations: \begin{equation} \begin{cases} x = \rho\sin\phi\cos\theta \\ y = \rho\sin\phi\sin\theta \\ z = \rho\cos\phi \end{cases} . \end{equation}

Vectors

A vector can be considered geometrically as a displace between two points, say from point $P$ to point $Q$, and it is represented as an arrow with its tail at $P$ and the tip at $Q$. The vector would be based at $P$, and $P$ would be called the "base point'' of the vector. Algebraically, a vector is described by its base point and its displacements from the base point parallel to the axes of a Cartesian co-ordinate system. These displacements are called the "components'' of the vector. As an example, consider the point $P=(1,2,2)$, based here might be a vector in $\Rthree$, $\vect{v} = (2,1,-2)_P$. Here, the tail of the arrow would be at $P$, and the tip of vector would be at the point $(3,3,0)$.

Definition (Vector addition) If $\vect{a} = (a_1,a_2)_P$ and $\vect{b} = (b_1,b_2)_P$ are two vectors in $\Rtwo$ based at the same point $P$, then, \begin{equation} \vect{a} + \vect{b} = (a_1 + b_1, a_2 + b_2)_P . \end{equation} Similarly in $\Rthree$, if $\vect{a} = (a_1,a_2,a_3)_P$ and $\vect{b} = (b_1,b_2,b_3)_P$ (here, $P$ is a point in $\Rthree$), then \begin{equation} \vect{a} + \vect{b} = (a_1 + b_1, a_2 + b_2, a_3 + b_3)_P . \end{equation} Geometrically, one can consider the two vectors as being adjacent sides of a parallelogram, then the sum of the two vectors is the vector that lies along the diagonal of the parallelogram.

Example \begin{align} \mathrm{2D: }& (-1,3)_{(2,1)} + (5,-4)_{(2,1)} = (4,-1)_{(2,1)} \\ \mathrm{3D: }& (2,1,4)_{(1,1,0)} + (1,-6,0)_{(1,1,0)} = (3,-5,4)_{(1,1,0)} \end{align}

Definition (Scalar multiplication) If $\vect{a} = (a_1, a_2)_P$ is a vector in $\Rtwo$, and $\alpha \in \R$, then $\alpha\vect{a}$ is the vector \begin{equation} \alpha\vect{a} = (\alpha a_1, \alpha a_2)_P .\end{equation} Similarly, in three dimensions, if $\vect{a} = (a_1,a_2,a_3)_P$ (where $P$ is a point in $\Rthree$), then $\alpha\vect{a}$ is the vector \begin{equation} \alpha\vect{a} = (\alpha a_1, \alpha a_2, \alpha a_3)_P .\end{equation}

Example \begin{align} \mathrm{2D: }& \sqrt{2}\left(\sqrt{5},\sqrt{18}\right)_{(1,2)} = \left(\sqrt{10},6\right)_{(1,2)} \\ \mathrm{3D: }& (-3)\left(1,2,\sqrt{2}\right)_{(1,2,1)} = \left(-3,-6,-3\sqrt{2}\right)_{(1,2,1)} \end{align} It is important to note that since the addition of vectors and the multiplication of vectors by scalars involves operations on the components of the vectors, which are all elements of $\R$, the other rules of algebra follow.

Theorem (Properties of addition and scalar multiplication) Let $P$ be a point (in either $\Rtwo$ or $\Rthree$), and let $\V_P$ denote the set of all vectors based at $P$. Then, if $\vect{a}$ and $\vect{b}$ are in $\V_P$, so is their sum, $\vect{a} + \vect{b}$ (closure over vector addition). Also, if $\vect{a}$ is in $\V_P$ and $\alpha$ is in $\R$, the product $\alpha\vect{a}$ is in $\V_P$ (closure over scalar multiplication). The following properties are satisfied:

• (Associativity of vector addition) For every $\vect{a}, \vect{b}, \vect{c} \in \V_P$, \begin{equation}(\vect{a} + \vect{b}) + \vect{c} = \vect{a} + (\vect{b} + \vect{c}) .\end{equation}
• (Vector additive identity) There is a zero vector in $\V_P$, denoted $\vect{0}_P$, such that \begin{equation}\vect{0}_P + \vect{a} = \vect{a} + \vect{0}_P = \vect{a} ,\end{equation} for every vector $\vect{a}$ in $\V_P$.
• (Vector additive inverse) For every $\vect{a} \in \V_P$, there exists a vector $-\vect{a} \in \V_P$, such that \begin{equation}\vect{a} + (-\vect{a}) = (-\vect{a}) + \vect{a} = \vect{0}_P .\end{equation}
• (Commutativity of vector addition) For every $\vect{a}, \vect{b} \in \V_P$, \begin{equation} \vect{a} + \vect{b} = \vect{b} + \vect{a} .\end{equation}
• (Distributivity of scalar multiplication of vectors) For every $\vect{a}, \vect{b} \in \V_P$, and every $\alpha \in \R$, \begin{equation}\alpha(\vect{a} + \vect{b}) = \alpha\vect{a} + \alpha\vect{b} .\end{equation}
• (Distributivity of scalar multiplication of vectors) For every $\vect{a} \in \V_P$, and every $\alpha,\beta \in \R$, \begin{equation}(\alpha + \beta)\vect{a} = \alpha\vect{a} + \beta\vect{a} .\end{equation}
• (Associativity of scalar multiplication of vectors) For every $\vect{a} \in \V_P$, and every $\alpha,\beta \in \R$, \begin{equation}(\alpha\beta)\vect{a} = \alpha(\beta\vect{a}) .\end{equation}
• (Scalar multiplicative identity of vectors) For every $\vect{a} \in \V_P$, \begin{equation}(1)\vect{a} = \vect{a} .\end{equation} (Here, 1 is the real number 'one'.)

Vector Spaces

The properties listed above for vectors turn out to apply in other contexts as well, and the notion of what a "vector space'' might be can be generalized.

Definition A vector space is a set \V of objects, together with operations of addition and scalar multiplication on these objects, such that the sum of two elements of \V is also an element of \V; the product of a real number and an element of \V is also an element of \V; finally, the eight properties listed in the theorem above also apply. (The elements of \V are then called vectors.) This definition allows for one to propose any set as a vector space, if one also provides the rules for addition of two elements of the set as well as the rule for how scalars multiply with elements of the set. These rules are usually self-evident but for more abstract elements, they may be necessary. Such a proposed set then may be verified as truly being a vector space or not by testing whether or not the conditions elaborated in the definition of a vector space above have been satisfied.

Example Let \V be the set of all functions on the closed interval $[0,1]$. We now need to have definitions for how elements in \V are added, and for scalar multiplication of elements in \V. Since the elements in \V are functions, we already know these operations: for every $f$ and $g$ in \V, and $\alpha\in\R$, $$(f+g)(x) = f(x) + g(x) \text{ and } (\alpha f)(x) = \alpha \cdot f(x) .$$ We see that both $f+g$ and $\alpha f$ are still in \V, and one can further verify every property in the theorem above, thus \V is a vector space.

Vector Operations

Beyond vector addition and scalar multiplication, vectors have other operations available.

The Vector Norm

The norm of a vector is the length of a vector. Recall that we had introduced the notion of a vector as being a displacement from one point to another. So, if a vector $\vect{v}$ is the displacement from point $P$ to the point $Q$, the norm of $\vect{v}$ is the distance from $P$ to $Q$. Thus, the norm of $\vect{v}$ may be written by using the distance formula.

Definition The norm of a two-dimensional vector $\vect{a}=(a_1,a_2)_P$ is given by \begin{equation}\norm{\vect{a}} := \sqrt{a_1^2 + a_2^2} .\end{equation} Similarly, for a three-dimensional vector $\vect{b} = (b_1, b_2, b_3)_P$, the norm is \begin{equation}\norm{\vect{b}} := \sqrt{b_1^2 + b_2^2 + b_3^2} .\end{equation} A vector with norm equal to one is called a unit vector. Note that the norm operation is defined using double bars. This is to indicate that the norm operation is similar in nature to the absolute value, but it is not the absolute value. We will not assign any meaning to the notion of the absolute value of a vector.

Example \begin{align} \mathrm{2D: }& \norm{(3,1)_{(1,1)}} = \sqrt{3^2 + 1^2} = \sqrt{10}\\ \mathrm{3D: }& \norm{(1,2,2)_{(3,1,1)}} = \sqrt{1^2 + 2^2 + 2^2} = 3 \end{align}

The Dot Product

The dot product, also called the scalar product, or the inner product, is an operation that maps two vectors to a real number.

Definition If $\vect{a} = (a_1, a_2)_P$ and $\vect{b} = (b1, b2)_P$ are two-dimensional vectors based at the same point, their dot product is the real number given by \begin{equation} \vect{a}\cdot\vect{b} := a_1b_1 + a_2b_2 .\end{equation} Similarly, in three dimensions, for $\vect{a} = (a_1, a_2, a_3)_P$ and $\vect{b} = (b1, b2, b_3)_P$, the dot product is \begin{equation} \vect{a}\cdot\vect{b} := a_1b_1 + a_2b_2 + a_3b_3 .\end{equation}

Example \begin{align} \mathrm{2D: }& (3,1)_{(1,1)} \cdot (2,3)_{(1,1)} = (3)(2) + (1)(3) = 9\\ \mathrm{3D: }& (1,2,0)_{(1,1,2)} \cdot (2,3,-1)_{(1,1,2)} = (1)(2) + (2)(3) + (0)(-1) = 8 \end{align} Geometrically, the dot product is proportional to the projection of one vector along the axis of the other vector. This means that for vectors $\vect{a}$ and $\vect{b}$ based at the same point, if a right triangle is created by drawing a line from the tip of one vector towards the other vector (so that it is perpendicularly incident), then $\vect{a}\cdot\vect{b}$ is the length of the side of the right triangle along the other vector times the length of that side. This interpretation gives the following formula for the dot product: \begin{equation} \vect{a} \cdot \vect{b} = \norm{\vect{a}} \norm{\vect{b}} \cos\theta ,\end{equation} where, $\theta$ is the angle between $\vect{a}$ and $\vect{b}$. Note that this formula relates the norm operation to the dot product, since \begin{equation} \norm{\vect{a}}^2 = \vect{a}\cdot\vect{a} , \end{equation} since, the angle between the vector $\vect{a}$ and itself is of course zero, and the cosine of zero is one.

Example Find the angle between the vectors $(1,0,1)_P$ and $(0,1,1)_P$. \\ We first calculate the norms of the two vectors: $$\norm{(1,0,1)_P} = \sqrt{2} \text{ and also } \norm{(0,1,1)_P} = \sqrt{2} .$$ So, $$(1,0,1)_P \cdot (0,1,1)_P = \sqrt{2}\sqrt{2}\cos\theta ,$$ but, can calculate $(1,0,1)_P \cdot (0,1,1)_P$ directly: $$(1,0,1)_P \cdot (0,1,1)_P = (1)(0) + (0)(1) + (1)(1) = 1 .$$ Therefore, $\theta = \arccos\frac{1}{2} = \frac{\pi}{3}$.

Definition Vectors $\vect{a}$ and $\vect{b}$, based at the same point, are said to be orthogonal if $\vect{a}\cdot\vect{b} = 0$. Note that if two vectors are orthogonal, they need not be perpendicular. Vectors of course are perpendicular if the angle between them is $\frac{\pi}{2}$. Meanwhile, if the dot product is equal to zero, this means that either the two vectors are perpendicular or at least one of the two vectors has a zero norm.

Theorem (Properties of the norm and dot product) Let $P$ be a point in $\Rtwo$ or $\Rthree$. Let $\vect{a}$, $\vect{b}$, and $\vect{c}$ be vectors based at $P$, and let $\alpha$ be a real number. Then,

• (Commutativity of the dot product) $\vect{a} \cdot \vect{b} = \vect{b} \cdot \vect{a}$
• (Distributivity of the dot product) $(\vect{a} + \vect{b}) \cdot \vect{c} = \vect{a}\cdot\vect{c} + \vect{b}\cdot\vect{c}$
• (Distributivity of the dot product) $\vect{a} \cdot (\vect{b} + \vect{c}) = \vect{a}\cdot\vect{b} + \vect{a}\cdot\vect{c}$
• (Associativity of the dot product) $(\alpha\vect{a})\cdot\vect{b} = \alpha(\vect{a}\cdot\vect{b}) = \vect{a}\cdot(\alpha\vect{b})$
• (Relation of the norm and dot product) $\vect{a}\cdot\vect{a} = \norm{\vect{a}}^2$
• (Norm of the scalar multiple of a vector) $\norm{\alpha\vect{a}} = \abs{\alpha}\norm{\vect{a}}$
• $\vect{a}\cdot\vect{b} = \norm{\vect{a}} \norm{\vect{b}} \cos\theta$, where $\theta$ is the angle between \vect{a} and \vect{b}.
• (Schwarz's inequality) $\abs{\vect{a}\cdot\vect{b}} \le \norm{\vect{a}} \norm{\vect{b}}$
• (The triangle inequality) $\norm{\vect{a} + \vect{b}} \le \norm{\vect{a}} + \norm{\vect{b}}$

The Cross Product

In the last section, we described the dot product, an operation that maps two vectors to a number. We now introduce the cross product, an operation that maps two vectors to another vector. Another difference to note is that while the dot product is defined in both two and three dimensions (actually in any number of dimensions), the cross product is only defined in three dimensions. (Actually, it happens to also be defined in seven dimensions.) The definition of the cross product is motivated as a method of finding a vector that is orthogonal to two other given vectors.

Definition Let $\vect{a} = (a_1, a_2, a_3)_P$ and $\vect{b} = (b_1, b_2, b_3)_P$ be two three-dimensional vectors based at the same point. The cross product of $\vect{a}$ and $\vect{b}$ (in that order) is the vector \begin{equation} \vect{a}\times\vect{b} := (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)_P . \end{equation}

Example Let $\vect{a} = (2,-1,3)_{(1,1,1)}$ and $\vect{b} = (-1,3,2)_{(1,1,1)}$. Then, $$\vect{a}\times\vect{b} = \left((-1)(2) - (3)(3), (3)(-1) - (2)(2), (2)(3) - (-1)(-1)\right)_P = (-11, -7, 5)_P .$$

Theorem Let $P$ be a point in $\Rthree$. Let $\vect{a}$, $\vect{b}$, and $\vect{c}$ be vectors based at $P$, and let $\alpha$ be a real number. Then,

• (Anti-commutativity of the cross product) $\vect{a}\times\vect{b} = -\vect{b}\times\vect{a}$
• (Distributivity of the cross product) $(\vect{a} + \vect{b})\times\vect{c} = \vect{a}\times\vect{c} + \vect{b}\times\vect{c}$
• (Distributivity of the cross product) $\vect{a}\times (\vect{b} + \vect{c}) = \vect{a}\times\vect{b} + \vect{a}\times\vect{c}$
• (Associativity of the cross product) $(\alpha\vect{a})\times\vect{b} = \alpha(\vect{a}\times\vect{b}) = \vect{a}\times(\alpha\vect{b})$
• (Norm of the cross product) $\norm{\vect{a}\times\vect{b}} = \norm{\vect{a}} \norm{\vect{b}} \abs{\sin\theta}$, where $\theta$ is the angle between $\vect{a}$ and $\vect{b}$
• $\vect{a}\times\vect{b} = \vect{0}_P$, if and only if $\vect{a}$ and $\vect{b}$ are parallel; $\vect{a}\times\vect{a} = \vect{0}_P$, for every vector $\vect{a}$

The cross product also has a geometric interpretation, which is very important. If two 3D vectors $\vect{a}$ and $\vect{b}$ are taken to be two adjacent sides of a parallelogram, then $\norm{\vect{a}\times\vect{b}}$ is the area of the parallelogram. (This also means that the triangle formed by the common base point of the two vectors and the two tips of the vectors has area $\frac{1}{2}\norm{\vect{a}\times\vect{b}}$.)

The Triple Scalar Product

The triple scalar product is a combination of the dot product and cross product that maps three vectors (in a given order) to a single number, and it has a very interesting geometric interpretation.

Definition Let $P$ be a point in $\Rthree$, and let $\vect{a}$, $\vect{b}$, and $\vect{c}$ be three-dimensional vectors based at $P$. Then the triple scalar product of these three vectors (in that order) is $\vect{a}\cdot(\vect{b}\times\vect{c})$. Geometrically, the absolute value of the triple scalar product of three vectors is the volume of the parallelepiped with the three vectors at one of its corners.

The "Usual'' Unit Vectors

The unit vectors are a useful algebraic representation of vectors, that can simplify long computations of dot products and cross products of vectors. In 2D, we have $\vect{i}$ and $\vect{j}$. In 3D, we also have $\vect{k}$. They are always based at the origin $O$. Explicitly, $\vect{i} = (1,0,0)_O$, $\vect{j} = (0,1,0)_O$, $\vect{k} = (0,0,1)_O$. (In 2D, the third components of $\vect{i}$ and $\vect{j}$ are omitted.) We have the following algebra for the dot product of unit vectors: \begin{align} \vect{i}\cdot\vect{i} &= \vect{j}\cdot\vect{j} = \vect{k}\cdot\vect{k} = 1 \\ \vect{i}\cdot\vect{j} &= \vect{i}\cdot\vect{k} = \vect{j}\cdot\vect{k} = 0 \end{align} (Remember that the order of vectors in the dot product can be reversed.) For the cross product of these vectors, we have the following: \begin{align} \vect{i}\times\vect{i} &= \vect{j}\times\vect{j} = \vect{k}\times\vect{k} = \vect{0}_P \\ \vect{i}\times\vect{j} = \vect{k} , &\vect{j}\times\vect{k} = \vect{i} \text{ , and } \vect{k}\times\vect{i} = \vect{j} \\ \vect{j}\times\vect{i} = -\vect{k} , &\vect{k}\times\vect{j} = -\vect{i} \text{ , and } \vect{i}\times\vect{k} = -\vect{j} \end{align} An easy way to remember the cross product rules is to first remember that the cross product of any vector with itself is always the zero vector. Otherwise, remember the following order: $\vect{i}, \vect{j}, \vect{k}, \vect{i}$. If the cross product involves any pairs of unit vectors in that (left to right) order, then the result is the positive version of the other unit vector, else it is the negative version.

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