Question

Calculus III: Lines, planes, and surfaces in space.

Answer 1

Find the parametric and symmetric equations of the line passing A(1, 1, 1,) and B(4, 0, -1)

Answer 2

\[V_{AB}=<3, -1, -2>\]

Answer 3

\[x=1+3t,\ y=1-t,\ z=1-2t\]

Answer 4

you have the parametric equation correct. the symmetric equation is obtained by solving each equation for t. By reflexivity, namely, t = t, you obtain your symmetric equation.

Answer 5

\[t=\frac{x-1}{3}=1-y=\frac{1-z}{2}\]

Answer 6

@khoala4pham What are other ways of notating the parametric equation I solved? We can write vectors as V = i + j + k, or V=<i, j, k>. Is there other ways of viewing parametric equations?

Answer 7

I can't immediately think of other ways of notation; however, your equations are sufficient. Utilizing vectors is very cool in that there is no one correct parametrization and what's more, vector calc emphasizes "independence of parametrization" which means that if the parametrizations express the same object, then the math on that object is the same regardless of how you parametrized the object.

Answer 8

@khoala4pham Ok. These parametric equations are a bit over my head, I understand how to solve for them going off of an example, but it is still not fully clicking with me. Thank you for your assistance, I will tag you in the next query if you are interested in helping us evaluate a similar problem.

Answer 9

I had trouble at first, too. Perhaps your problem is that the parametrizations are essentially the range of a function. For graphs of f(x) = x, you plot it in a plane (2 dimensions) where you plot both the domain and range.
The parametrization however is r(t) = <x(t), y(t), z(t)> and we are only concerned with the range. We don't plot t in our equations, just the x, y, and z. It'll take some time perhaps, but it's one of those things that you have to get used to.

Answer 10

@khoala4pham So does that mean that \[r(t)=<1+3t,\ 1-t,\ 1-2t>\]?

Answer 11

yes. That parametrization gives you the line. that passes through those two points in 3 space. What's more, is that if you restrict 0<t<1, it will be the line segment that connects those two points.
Another hidden point that mathematicians abuse is that r(t) is a position vector however, in parametrizing, r(t) is treated as the set of ENDPOINTS of such said vectors. It is a very fine point but mathematicians like to use it like that.