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Find parametric equations and symmetric equations for the line.

The line of intersection of the planes $ x + 2y + 3z = 1 $ and $ x - y + z = 1 $

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The parametric equation:$x=5 t \quad y=\frac{-2}{5}+2 t \quad z=\frac{3}{5}-3 t$The symmetric equation:$t=\frac{x}{5}=\frac{y+\frac{2}{5}}{2}=\frac{z-\frac{3}{5}}{-3}$

03:54

Wen Zheng

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 5

Equations of Lines and Planes

Vectors

Johns Hopkins University

Oregon State University

University of Michigan - Ann Arbor

Boston College

Lectures

02:56

In mathematics, a vector (from the Latin word "vehere" meaning "to carry") is a geometric entity that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. Vectors play an important role in physics, engineering, and mathematics.

11:08

In mathematics, a vector (from the Latin word "vehere" which means "to carry") is a geometric object that has a magnitude (or length) and direction. A vector can be thought of as an arrow in Euclidean space, drawn from the origin of the space to a point, and denoted by a letter. The magnitude of the vector is the distance from the origin to the point, and the direction is the angle between the direction of the vector and the axis, measured counterclockwise.

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So you can see that here. I have three unknowns and two equations. So I protect one variable is free. So let's take you to be to be sorry and gt then this becomes X plus I'm sorry this is X-plus two y Plus three T is equal to one. The next minus Y, Nasty goes to one, There's three D. So just take teach together side. So this is X plus two, Y is one minus treaty and the next man is why is one dynasty. We call this equation one we call the situation too. And if i subtract a question to from a question one question one minor situation too. Gibbs. What does it give? It gives us that extras to y minus X plus Y. So excess cancels out. This is T. Y Equals 1- Treaty minus one lusty. This is minor stupid. So why becomes -2/30. He was already. And then all I have to do is just tell you what access but I already know that X minus Y. Is one dynasty. So that tells us the taxes. Why? Last one dynasty Which is -2 over three D. Plus one dynasty. To solve this. This becomes 1 -5/3 people. So let's go to the other page. Okay I'm sorry let's stay here for a while. And then what we know now is that X is -5/3 T plus one lying is -2/30 and G is just this is not real. So this is my automatically question now to find the symmetric depression. This is already in the form of a parametric equation, there's nothing but nothing to do here. This is my parametric equation. Find the symmetric equation. So what we do is just we solve for T. So X -1. You hired by -5/3 must be equal to Y divided by minus 2/3. This must be called G over one and that's it, that is my symmetric aggression and that is the answer. Again, it depends on what your tickets of parameter. So the answer could be many answers and if you take extremity as well and it doesn't really matter.

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