gtMath

${\Bbb R}^{n}$

Imagine the number line floating in space, and at 0, we put another copy of the number line at a 90 degree angle to the first. These are the $x$- and $y$-axes, and together, they make a plane (a flat sheet that goes on forever in all directions). Any point on the plane can be identified by how far out it is on the $x$-axis and then how far out on the $y$-axis. The two numbers describing the point's position are called the point's coordinates, and the plane is called the $xy$-plane. The set of points on the number line, for example 4 or 3.78123947321 or $\sqrt{2} \approx 1.41421356237$, is denoted by the symbol ${\Bbb R}$. If you are not familiar with sets and the notations for describing them, check out the post on sets.

A point in the $xy$-plane with coordinates $x_0$ and $y_0$ is written as an ordered pair $(x_0,y_0)$. The set of ordered pairs of real numbers is the Cartesian Product of ${\Bbb R}$ with itself, i.e. ${\Bbb R} \times {\Bbb R}$ and is also written ${\Bbb R}^{2}$. So ${\Bbb R}^{2}$ is the plane.

Now imagine we add a third copy of the number line to our $xy$-plane at the point $(0,0)$ (i.e. where the $x$- and $y$-axes cross), this time perpendicular to the plane. This is the $z$-axis, and now we have a 3-dimensional space where a point is identified by an ordered triple $(x,y,z)$. The set of such points is called ${\Bbb R}^{3}$ and is just ordinary 3-dimensional space.



If $n$ is an integer greater than 3, then we could consider the set of ordered n-tuples $(x_1, x_2, x_3, ... , x_n)$. This is ${\Bbb R}^{n}$ or $n$-dimensional space. We can't picture this space, but we can do the same math with it that we can do with ${\Bbb R}^{2}$ and ${\Bbb R}^{3}$, so it has many applications. Let's focus on ${\Bbb R}^{3}$ for now.

The $x$-, $y$-, and $z$-axes divide 3-dimensional space into 8 regions, called octants. For example, one octant consists of points whose $x$-, $y$-, and $z$-coordinates are all positive. Another octant contains points whose $x$-coordinate is positive, but whose $y$- and $z$-coordinates are negative. And so on. There are 3 coordinates and 2 possibilities (positive or negative) for each coordinate, thus $2^3 = 8$ regions. Points with one coordinate whose value is equal to 0 lie in one of the coordinate planes ($xy$-plane, $xz$-plane, or $yz$-plane). Points with two 0 coordinates lie on one of the axes, and the point with all coordinates equal to 0 is called the origin.

The distance between two points $\textbf{x}_{1}=(x_1,y_1,z_1)$ and $\textbf{x}_{2}=(x_2,y_2,z_2)$ is the length of the straight line segment between them. The formula for this distance (the distance formula) is $d(\textbf{x}_{1},\textbf{x}_{2})=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$, which can be worked out by using the Pythagorean Theorem a few times. There could be alternative ways of measuring distance under which a straight line would not be the shortest distance between two points (e.g. measuring distance on the surface of a sphere which sits in ${\Bbb R}^{3}$), but let's stick with the straight-line distance. ${\Bbb R}^{3}$ endowed with this metric for distance is known as (3-dimensional) Euclidean space.

Vectors

A point in ${\Bbb R}^{2}$ or ${\Bbb R}^{3}$ can also be thought of as a vector, which is an arrow starting at the origin and terminating at the point in question. The origin is the zero vector, denoted $\vec{0}$ or $\bf{0}$. A vector has a magnitude, which is its length, and a direction in which it points. So for example, you could have a vector that is 30 degrees from the $x$-axis, 45 degrees from the $xy$-plane in the positive $z$ direction (i.e. up), and with length 5. This vector would terminate at the point $(\frac{5\sqrt{6}}{4},\frac{5\sqrt{2}}{4},\frac{5\sqrt{2}}{2}) \approx (3.06,1.77,3.54)$. Can you figure out how I got those coordinates from the description of the angles and length?



The 3 coordinates specify the direction (which can also be equated with angles from the axes using some sines and cosines), and the magnitude of the vector can be worked out using the distance formula. For a vector $\vec{x} = (x,y,z)$, the magnitude is $\|\vec{x}\| = \sqrt{x^{2}+y^{2}+z^{2}}$. This is a direct consequence of the distance formula mentioned above. Note that in addition to $\vec{x}$, we often see vectors written as $\bf{x}$ or $\vec{\bf{x}}$.

In physics, vectors are used to represent, among other things, an object's velocity, which has a magnitude (speed) and direction that changes over time.

Operations with Vectors

 

In the context of vectors of ${\Bbb R}^{n}$, a regular number $c \in {\Bbb R}$ is called a scalar (since it scales a vector).

Given a vector ${\bf x} = (x,y,z)$ (note that $x$, $y$, and $z$ are real numbers as stated above) and a scalar $c$, we can define a new vector $c {\bf x}$ by $c{\bf x} = (cx,cy,cz)$. This operation on a vector is called scalar multiplication. A vector whose magnitude is 1 is called a unit vector. To make a unit vector in the same direction as a vector ${\bf x}$, simply scalar multiply ${\bf x}$ by the number $\frac{1}{\| {\bf x} \|}$.

We can also add two vectors ${\bf x} = (x_1,x_2,x_3)$ and ${\bf y} = (y_1,y_2,y_3)$ by defining the new vector ${\bf x}+{\bf y}$ to be the vector whose coordinates are the sum of the coordinates of ${\bf x}$ and ${\bf y}$, i.e. ${\bf x}+{\bf y} = (x_1+y_1,x_2+y_2,x_3+y_3)$. This is called vector addition.



The scalar multiplication and vector addition defined above satisfy the following eight properties (which can be proved easily using the definitions) called the vector space axioms. In the following, boldface symbols represent vectors in ${\Bbb R}^{3}$ and italic symbols and non-boldface numbers represent scalars, i.e. elements of ${\Bbb R}$.

Associativity of vector addition: 
$\ \ \ \ \ \ \ \ {\bf x} + ({\bf y} + {\bf z}) = ({\bf x} + {\bf y}) +{\bf z}$
Commutativity of vector addition:
$\ \ \ \ \ \ \ \ {\bf x} + {\bf y} = {\bf y} + {\bf x}$
Additive identity: for all vectors ${\bf x}$, the zero vector ${\bf 0} = (0,0,0)$ has the property that $\ \ \ \ \ \ \ \ {\bf 0} + {\bf x} = {\bf x}$
Additive inverse: for a vector ${\bf x} = (x_1,x_2,x_3)$, the vector $-{\bf x} = (-x_1,-x_2,-x_3)$ has the property that
$\ \ \ \ \ \ \ \ -{\bf x} + {\bf x} = {\bf 0}$
Compatibility of scalar multiplication and field multiplication:
$\ \ \ \ \ \ \ \ a(b{\bf x}) = (ab){\bf x}$
Identity element of scalar multiplication:
$\ \ \ \ \ \ \ \ 1{\bf x} = {\bf x}$
Distributivity of scalar multiplication with respect to vector addition:
$\ \ \ \ \ \ \ \ a({\bf x} + {\bf y}) = a{\bf x} + a{\bf y}$
Distributivity of scalar multiplication with respect to field addition:
$\ \ \ \ \ \ \ \ (a+b){\bf x} = a{\bf x} + b{\bf x}$

Note: these properties inspire the definition of a vector space as any set (whose elements we call vectors) and a set of numbers (from a field, whose elements we call scalars, usually the real numbers ${\Bbb R}$ or the complex numbers ${\Bbb C}$) with a defintion of scalar multiplication and vector addition that satisfy the axioms above. For example, the set of functions $f$ mapping real numbers $x$ to real numbers $f(x)$ are a vector space with the real numbers as scalars if we define scalar multiplication by $(af)(x) = af(x)$ and vector addition by $(f+g)(x) = f(x) + g(x)$. You can prove that these definitions satisfy the axioms as an exercise if you want.

There are two more important vector operations that allow us to "multiply" two vectors in ${\Bbb R}^{3}$.

The dot product (also known as the scalar product or inner product) of two vectors produces a scalar: ${\bf x} \cdot {\bf y} = x_1 y_1 + x_2 y_2 + x_3 y_3$. This is also defined in ${\Bbb R}^{n}$ when $n>3$ in the same way, but with $n$ summands instead of 3: $${\bf x} \cdot {\bf y} = \sum_{i=1}^{n}{x_i y_i}$$ The dot product is related to the angle $\theta$ between two vectors by the formula ${\bf x} \cdot {\bf y} = \| {\bf x} \| \| {\bf y} \| \cos \theta$. So the dot product is maximized when the vectors point in the same direction, zero when they are perpendicular, and minimized (negative) when the vectors point in opposite directions.

The cross product is only defined in ${\Bbb R}^{3}$ and produces a new vector whose direction is perpendicular to both of the two input vectors and whose magnitude is equal to the area of the parallelogram spanned by the two vectors, which is $\| {\bf x} \| \| {\bf y} \| \sin \theta$. The definition of the cross product is ${\bf x} \times {\bf y} = (x_2 y_3 - y_2 x_3, - x_1 y_3 + x_3 y_1, x_1 y_2 - x_2 y_1)$.

To obtain a vector perpendicular (also known as normal) to two vectors, we can take their cross product, and we can always make it into a unit normal vector by dividing by (i.e. scalar multiplying by 1 divided by) the magnitude of the cross product.