Advanced Math/Linear algebra 1
QUESTION: I am having difficulty understanding what to do for these questions. Any help would be greatly appreciated.
What is the relationship of the given line
L: (x,y,z) = (-2,1,5) + t<2,-8,-6>
to each of the lines below? Is it coincident, parallel, intersecting or skew?
1. (1, -3, 1) + t<1,2,3>
2. (4,-5,-1) + t<-5, 20, 15>
3. (5,5,3) + t<3,-4,1>
4. (-2,11,21) + t<-2,3,-2>
5. (-1,-3,2) + t<-3,12,9>
ANSWER: Billy, Timmy, Sam,
Wish you had more specific questions or could at least list some of the definitions and results presented to you in your course work. The subject of lines and planes in 3-D is covered pretty succinctly in various references; one example that you should look at is
Just to get our terms straight,
paralell means the direction numbers, i.e., the vectors denoted by < >, are parallel, which again means the components are proportional (with the same proportionality constant);
coincident means the lines are both parallel and contain the same point;
intersect means that a single point is a soluton both the equations for the 2 lines;
skew means none of these conditions is met.
For instance, for the line given by (-2,1,5) + <2,-8,-6> and the line in 2. (4,-5,-1) + <-5,20,15>, the direction vectors are related by <2,-8,-6> = (-2/5)<-5,20,15>, i.e., they are proportional/parallel. Same is true for 5. (the proportionality constant is -2/3).
To determine if 2 lines that are not parallel still intersect, you must determine if a point exists which solves the equations for the 2 lines. For the given line and line 3, we have
(-2,1,5) + t<2,-8,-6> = (5,5,3) + s<3,-4,1> for some values of t and s (note that I have substituted s for t in the second equation to make it clear that the parameters are independent for the 2 lines). Equating the x,y and z components gives
x: -2+2t = 5+3s
y: 1-8t = 5-4s
z: 5-6t = 3+s.
Can these be solved for specific values of x, y, z? To find out, use the x and y equations to solve for unique values of t and s and then substitute these values into the z equation to see if it holds. If it does, then the lines intersect. My arithmetic shows that for this example, there is no solution. (therefore they are skewed).
You can use this method to test the other equations. Let me know what your results are.
---------- FOLLOW-UP ----------
QUESTION: I got these as my answers:
It is telling me one of the answers is wrong but i dont know which one is.
I get that line #4 intersects L. The direction numbers are not proportional so they are not parallel. But solving for s and t using the x and y eqns, I get s = 2, t = -2. Plugging these values into the z eqn gives 17 = 17, so that the lines intersect. The point of intersection, given by L by using t=-2 is (-4,9,11).