Lines and Planes in Space
Professor Leonard – Calculus 3 – 11.5 ‘Lines and Planes in 3D’
Thomas – 12.5 ‘Lines and Planes in Space’
If we have a point on an unknown line L, and we also have a vector v with which the unknown line is parallel, then we can construct the equation for L. We know from basic geometry that given two points, (x, y), on a 2D plane, that we can draw a straight line and we can describe the slope of that line (rise/run). Correspondingly, if we know one point and know the slope, we can describe the line. Furthermore, if we want to draw a line segment, we define the length as a multiple (a scalar) of the slope that terminates at our given point.
Right, that’s all review. In 3-dimensional space, we can chart lines by adding an additional coordinate. For our purposes, we’ll say that we go from (x, y) to (x, y , z). Given two (x, y, z) coordinates, we can form a line in 3D space. We’ll use direction vectors <x, y, z> to describe the, well, direction of the line is that space.
So, a line L in 3D Space can be defined like this: P1<x, y, z> – P2<x, y, z>, or <x2 – x1, y2, – y1, z2 – z1>. Alternatively, we could say that L = direction vector v *t where t is a scalar that augments the length of L. Often, t represents time. This allows us to think about the movement of things in given directions, outlined by their direction vectors, over a period of time.
This is all fine and dandy when we’re working with straight lines. But as is often the case in Calculus, as in our day to day lives, movements and trends arc, bend, twist, and wiggle around in all sorts of non-straight manners. To describe these lines, we use parametric equations which more accurately track the individual movements over each coordinate, like this:
t = (x2 – x1)/a, t = (y2 – y1)/b, t = (z2 – z1)/c
Here we have our t variable, the delta over that t, and the a, b, and c coordinates that make up our direction vector <a, b, c>.
Describing lines in this way gives us much greater flexibility and range of operations. By describing lines in terms of parametric equations, we can find out if lines are parallels (if they’re scalar multiples of each other), or if they intersect (they pass through the same point). Of course, this can do much for us. With two intersecting lines we can form parallelograms. Moreover, if we know a point and a vector (the normal vector) which is perpendicular to a heretofore unknown plane, then we can describe the plane. Let’s say we have:
Point P(x, y, z) and normal vector v = <a, b, c>, then the plane can be described thus:
ax + by + cz = d
Follow this along, now we can determine whether planes are parallel or perpendicular. We can also find the angle between planes
It’s beautiful how math comes together like that!
This section was actually quite large. Professor Leonard’s lecture was nearly 3.5 hours long! There will be much I’ll need to drop back to and review as I move forward. Fortunately, none of the concepts on their own were particularly mind-bending. It’s all just a matter of practice. And writing. Until later…