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# Parametric representations of lines

Parametric Representations of Lines in R2 and R3. Created by Sal Khan.

## Want to join the conversation?

• How would you use this in real life? •   You may want to describe the motion of a particule, say an electron. Then you need the representation of this path. Think about the parameter t as the time.
• How can a line be represented in 50 dimensions?What are its real-time applications?I mean, what is the need of a line represented in 50 dimensions. •   You can't graphically represent a line in 50 dimensions, because we only have three spacial dimensions to work with. A line in 50 dimensions would just be a representation of a set of values. Think of it, like this: In two dimensions I can solve for a specific point on a function or I can represent the function itself via an equation (i.e. a line). In three dimensions I can represent a point on a function or a line of a function or the function itself (a plane). So, in 50 dimensions a line would represent a range of specific values of a function.

Real-world applications would involve 50 different variables, e.g.: quantity of flavours of ice-cream (Strawberry, Chocolate, Vanilla, Blueberry, ... etc.) and the cost of an amount of a specific flavour. Then a line could be a representation of an optimal set of values for the quantity of ice-cream for each flavour at the lowest cost.

I hope that helps. (Please correct me if I'm wrong)
• What is collinear? •  "collinear" = "In the same line"
For example,
Any two points are colinear (because you can draw a line between them),
Two opposite points in a circle and its center are colinear.

Colinearity is interesting with three or more points.
• (13m:43s) Can we instead of doing b-a do a-b? • When determining the equation for L he says that it doesn't matter whether or not you were to use P1+t[P2-P1] or P2+t[P1-P2] or any other variation, or this is at least how I understood it. How is this possible if each variation provides you with different parametric equations? I attempted to apply this method to a problem from my calc txtbook and i found that the answer from the back of the book only corresponds with one of the forms of the equation. • Does your textbook say to find "a" set of parametric equations, or "the" set? Parametrizations for a given line are not unique, so your book should say "answers may vary", or something. Or, the problem would have to give you more information than the two points. If you look at t as a time parameter (you don't have to), you can think of different parametrizations of the same line as two particles going perhaps in opposite directions, or different speeds, but along the same path.
• How would someone reading the set S = {c v | c ( R} know that it represents only the line in R^2 going through the point (0, 0)?

Earlier on, we defined vectors such that v can be started anywhere on the Cartesian plane , not just the origin. Therefore, the set S should represent all lines in R^2 with the same slope as vector v .

I know that at , Sal makes the clarification that this is only if we define v as the position vector version of v, which starts only at the origin. But when writing the set S, is there a way (a notation of some sort) to clarify that c v represents only the set of all scaled position vectors v , rather than just all v ? Because otherwise, adding the x vector later on would make literally no difference to S. You'd just be taking an infinite number of lines in R^2 and moving all of them by x, which would have no visible effect. • Wait a minute...could you actually describe the path of a fly with a parametric equation, if the path were not a straight line? Or was he just referring to straight lines? • I think the confusion with the non-straight lines (or at least my confusion with it) derived from the fact that in the video, sal was showing us parameterization with just a straight line. The functions in this video always produced straight lines because they were of the form mx + b. It does not have to stay in that form though. You could make a function like x= t^3 + 3, y= t^2 , z= t^.5 + 2t. Here is a graph of that function I just stated to visualize it: http://www.wolframalpha.com/input/?i=plot+%7Bx+%3Dt%5E3+%2B+3%2C+y%3Dt%5E2%2C+z%3Dt%5E.5+%2B+2t%7D
• What is an example of higher dimensional problems in real life? • This may be a silly question, but at why B-A? Is not A+B the same vector? What am I missing? • A + B is not the same as B - A. You can see this clearly with the following example: Let A = [1,2], and B = [3,4]. B - A is [2,2] and A + B is [4,6].

A + B is the vector you get by drawing A, then drawing B with B's tail at the head/tip/front of A. My suggestion is to draw some actual vectors on some axes and give it a go.

A + B is easy to see, but B - A isn't as easy. Another way to think of it (hope this is not more confusing!) is that to get from A to B you need to add another vector. What is that vector? It is B - A. Let's see that in algebra:

A + (B - A)
= A + B - A
= B 