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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?

• Towards the end of the video Sal comments that in R3, R4...Rn spaces - we need a parametric equation to define a line. Is there a reason why? Is it becuase we introducing a 4th unknown to solve for t and thus reducing a plane (e.g. x+y+z=1) to a line? If this is correct, what happens if we introduce 2 unknowns (e.g. t+u) into a parametric equation? Do we get a point? •   For a system of m equations in n unknowns, where n >= (greater than or equal to) m, the solution will form an (n - m)space. So for one equation with one unknown like x = 7, the solution is a 0-space (a single point). For one equation in two unknowns like x + y = 7, the solution will be a (2 - 1 = 1)space (a line). For one equation in 3 unknowns like x + y + z = 7, the solution will be a 2-space (a plane).
For a system of parametric equations, this holds true as well. By strategically adding a new unknown, t, and breaking up the other unknowns into individual equations so that they each vary with regard only to t, the system then becomes n equations in n + 1 unknowns. The solution to this system forms an [(n + 1) - n = 1]space (a line).
If another unknown were added to the mix, the solution would actually end up being a 2-space (a plane).
• •  by getting a high paying job. 8)

This stuff is incredibly powerful and plays a role in almost anything that uses math(science, engineering, finance, economics, statistics). the scalar parameter could be time but also path length and and general coordinate - temperature, or distance along a beam or energy.
• 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)
• •  "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.
• • 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.
• I'm looking for info on dilation of a triangle with a scale factor. Could someone point me to that video if it exists, please? • 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.   