Subsection 2.2.1 Three-Dimensional Coordinate Systems
To specify the location of a point in our three dimensional world, three numbers are needed. For example, the position of an aircraft is specified by its latitude, longitude and altitude, which is to say by measuring distances east or west, north or south and above or below some reference point.
In mathematical terms, we can think of starting with a reference plane (mimicing the surface of the earth in a region small enough that the earth is close enough to flat), marking this plane with a reference point as origin and a grid measuring how far east or west (let us call this number the \(x\)-coordinate: positive for points east of the origin, negative for west) and how far north or south (let us call this number the \(y\)-coordinate: positive for points north of the origin, negative for south) a point in to plane is from this origin.
In a room with rectangular floor and walls you can specify the location of any point \(P\) in the room with three numbers; for example using one corner of the room as the reference point or origin.
First choose a corner as origin \(O\text{,}\) and name the three edges running from that corner as the \(x\text{,}\) \(y\) and \(z\) axes. (A common choice is putting the origin in the bottom-south-west corner and then using \(x\) for the edge running east, \(y\) for that running north, and \(z\) for that running upwards.) Thus the floor connects the \(x\) and \(y\) axes (the \(x\)-\(y\) plane), one wall meeting the origin ("south") connects the \(x\) and \(z\) axes (the \(x\)-\(z\) plane), and the other wall meeting the origin ("west") connects the \(y\) and \(z\) axes (the \(y\)-\(z\) plane).
The distance \(a\) horizontally west from the point to the "west wall" or \(y\)-\(z\) plane, along a line parallel to the \(x\) axis, is the \(x\)-coordinate; the values \(b\) and \(c\) of the \(y\)- and \(z\)-coordinates can be measured similarly.
How the coordinates determine the location of point \(P\).
How do these three coordinate values \(a\text{,}\) \(b\) and \(c\) determine the point \(P\text{,}\) and why does the origin have that name? To get to \(P\text{:}\)
start originally at the origin \(O\text{,}\)
move along the \(x\)-axis ("east") by a distance \(a\text{,}\)
then move parallel to the \(y\) axis ("north") across the \(x\)-\(y\) plane by a distance \(b\text{,}\)
and finally move parallel to the \(z\)-axis ("up") by a distance \(c\text{.}\)
Thus, a point in space can be described by a triple of numbers.
The point \(P\) with the above coordinate values is usually written \((a,b,c)\text{,}\) or to indicate the name too, \(P(a,b,c)\text{.}\)
The set of all such triples is denoted as \(\mathbb{R} \times \mathbb{R} \times \mathbb{R}\) or \(\mathbb{R}^3\text{.}\)
Exercise 2.2.1.
What surfaces in the 3D space \(\mathbb{R}^3\) are given by the following equations?
\(\displaystyle z=3\)
\(\displaystyle y=5\)
\(\displaystyle x+y=1\)
(Note that it is important to specify that we are in \(\mathbb{R}^3\text{,}\) especially in (b).)
Exercise 2.2.2.
Describe and sketch the surface in \(\mathbb{R}^3\) given by the equation \(y=x\text{.}\)
Exercise 2.2.3.
Find an equation for the sphere of radius \(r\text{,}\) center \(C(h,k,l)\text{.}\)
Exercise 2.2.4.
Show that \(x^2+y^2+z^2+4x-6y+2z+6=0\) is the equation of a sphere.
Which sphere?
Solid regions in space described by inequalities.
Much as an inequality (or several) in the two variables
\(x\) and
\(y\) describe a region within the
\(x\)-
\(y\) plane, inequalities in three variables describes solid regions in space. A simple example is that
\(x^2+y^2+z^2 \leq 1\) describes the solid ball of radius 1, center the origin.
Example 2.2.5.
Describe the region (in \(\mathbb{R}^3\)) given by the points whose coordinates satisfy the inequalities
\begin{equation*}
1 \leq x^2+y^2+z^2 \leq 4, \qquad z \leq 0.
\end{equation*}
Subsection 2.2.2 Vectors in \(\mathbb{R}^3\)
Much about vectors in
\(\mathbb{R}^3\) is very similar to what was seen in
Section 2.1, these notes will be brief, and focus on the more significant differences.
A vector \(\vec{v} = \veclong{PQ}\) in three dimensions describes the dispacement between two points \(P(p_1, p_2, p_3)\) and \(Q(q_1, q_2, q_3)\text{,}\) so that \(\vec{v} = \vector{v_1, v_2, v_3} = \vector{q_1-p_1, q_2-p_2, q_3-p_3}\text{,}\) and is the same so long as the displacement is the same; in particular, the above \(\vec{v}\) is also \(\veclong{OV}\) for \(O(0, 0, 0)\) the origin in \(\mathbb{R}^3\) and \(V(v_1, v_2, v_3)\text{.}\)
The rules for multiplication by a scalar (number), addition, and thus subtraction are as expected; for any \(\vec{v} = \vector{v_1, v_2, v_3}\text{,}\) \(\vec{w} = \vector{w_1, w_2, w_3}\) and scalar \(k \in \mathbb{R}\text{,}\)
\begin{equation*}
\begin{split}
k\vec{v} \amp= \vector{k v_1, k v_2, k v_3}
\\
\vec{v} \pm \vec{w} \amp= \vector{v_1 \pm w_1, v_2 \pm w_2, v_3 \pm w_3}
\end{split}
\end{equation*}
and these follow all the expected rules about commutativity, associativity and distributivity of multiplication over addition and subtraction, with the zero vector \(\vec{0} = \vector{0, 0, 0}\text{.}\)
Also, the length of a vector \(\vec{v} = \veclong{PQ}\) is the distance betwen the points \(P\) and \(Q\text{,}\) denoted either \(|PQ|\text{,}\) \(\|\vec{v}\|\) or (lazily) \(|\vec{v}|\text{:}\)
\begin{equation*}
\|\vec{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2}
\end{equation*}
With this, we can get a unit vector
\(\vec{u}\) in the same direction as a given non-zero vector:
\begin{equation*}
\|\vec{u}\| = \frac{1}{\|\vec{v}\|}\|\vec{v}\| \text{, also denoted } \hat{u}
\end{equation*}
because unit vectors are sometimes denoted with a "hat" instead of an arrow.
In particular, there are 3D versions of the standard unit vectors seen for vectors in the plane:
\begin{equation*}
\begin{split}
\vec{\imath} = \hat{\imath} = \vector{1, 0, 0}\\
\vec{\jmath} = \hat{\jmath} = \vector{0, 1, 0}\\
\vec{k} = \hat{k} = \vector{0, 0, 1}
\end{split}
\end{equation*}
Let \(\vecv = \vector{v_1, v_2, v_3}\) and \(\vecw = \vector{w_1, w_2, v_3}\) be vectors, and let \(k\) be a scalar; then:
Scalar multiplication: \(k\vecv = \vector{k v_1, k v_2,k v_3}\)
Vector addition: \(\vecv + \vecw = \vector{v_1+w_1, v_2+w_2,v_3+w_3}\)
Vector subtraction: \(\vecv - \vecw = \vector{v_1-w_1, v_2-w_2,v_3-w_3}\)
Vector magnitude: \(\|v\| = \sqrt{v_1^2+v_2^2+v_3^2}\)
Unit vector in the direction of \(\vecv\text{:}\) \(\hat{v} = \frac{1}{\|\vecv\|} \vecv =\vector{\frac{v_1}{\|\vecv\|}, \frac{v_2}{\|\vecv\|}, \frac{v_3}{\|\vecv\|}}\text{,}\) if \(\vecv \neq \vec{0}\text{.}\)
