5. Equations of Lines and Planes
Part 1: Plotting Lines Using Vectors
We are going to learn how to use the vector equation of a line to plot lines in Geogebra!
Recall that to find the equation for a line, we need two pieces of information:
(1) A point on the line, and (2) the direction of the line.
For lines in the typical Two Dimensional Cartesian plane, the direction of the line was always given by the slope or an angle of inclination and used points such as one or both of the points of intercept with the axes.
For higher dimensional spaces (in this case Three-Dimensional) we still only need a point on the line, denoted by the position vector and a vector,, pointing the direction of the line.
Then, as was derived in the lecture video, we get the following vector equation for such a line:
where is just a parameter that varies over all .
Instructions:
So, we want to construct such a line in Geogebra.
Let us begin by choosing a nice point on the line such as the point (0,0,2).
In the graphic below, construct the vector . (Hint: Geogebra lets you assign sub-indexed variables using the underscore(_) after the variable letter.)
Next, we will need a vector describing the direction of our line.
Again, in the graphic below, construct the vector .
Finally, it is time to construct the line!
In the input area type the following:
r = r_0 + t*v and hit enter.
Notice that Geogebra treats the letter t as an automatic parameter and fills out the entire line for our window.
One comment that is made in the textbook, is that one can think of the line as being "traced out" in space by the vector as varies.
We can see this visually by doing a little more to the graphic below!
In the next input area, type s and hit enter. You will see a slider appear on the left-hand side.
We are going to use this slider for our next trick!
Type the following: l = r_0 + s*v and hit enter.
This will result in a new vector that you can move using the slider. Notice that as you move the slider, the tip of traces out the line we drew previously!Part 2: Plotting Planes
Recall that to define a plane we need:
(1)a point on the plane, and (2) a vector normal to the plane.
Letting the point on the plane be given by and the vector normal to the plane by given by , we get the following equation defining the plane
or written another way
However, neither of these lead us to an 'easy-to-use' formula in Geogebra.
Nonetheless, we CAN use this, as is shown in the lecture, to derive the linear equation of a plane
where .
Using this linear form of the equation for the plane allows us to plot these planes in Geogebra (we may just have to do a little work on paper, first!)
For example, in the following graph type
2x + 2y + 2z + 2 = 0 and hit enter.
Geogebra will have plotted the plane corresponding to that linear equation. Note that you could just as well have entered 2x + 2y + 2z = -2 and, since these equations are equivalent, you would have gotten the same plane.
Go ahead and try it out with some other planes.
The trick of using the linear equation is extremely handy. We will use this method more in the next section when we talk about quadric surfaces.