Work out the line equation (parametric equation)

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In summary, the problem involves finding the equation of a line, r, that is the intersection of two planes, π1 and π2. The planes are defined by point A and non-coplanar vectors v1, v2, and v3, and any non-colinear vector u, respectively. The solution requires finding normal vectors, n1 and n2, to the two planes and considering the orientation of the line of intersection with respect to these normal vectors.
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Hernaner28
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Homework Statement


Let the point A and v1,v2,v3 non-coplanar vectors.
Let the plane
[tex]{\pi _1})P=A+{\lambda _1}{v_1}+{\lambda _2}{v_2}[/tex]

Consider any vector u non-colinear with v3 and the plane:
[tex]{\pi _2})P=A+{\lambda _3}{v_3}+{\lambda _4}u[/tex]

2. Task
Work out the equation of the line
[tex]r={\pi _1}\cap {\pi _2}[/tex]

The Attempt at a Solution


No idea how to start. I just know that the point A belongs to both planes and the point A belongs to a line with the equation form of: [tex]A+\lambda v[/tex] but I don't know what to do next.

Thanks!
 
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Hernaner28 said:

Homework Statement


Let the point A and v1,v2,v3 non-coplanar vectors.
Let the plane
[tex]{\pi _1})P=A+{\lambda _1}{v_1}+{\lambda _2}{v_2}[/tex]

Consider any vector u non-colinear with v3 and the plane:
[tex]{\pi _2})P=A+{\lambda _3}{v_3}+{\lambda _4}u[/tex]

2. Task
Work out the equation of the line
[tex]r={\pi _1}\cap {\pi _2}[/tex]

The Attempt at a Solution


No idea how to start. I just know that the point A belongs to both planes and the point A belongs to a line with the equation form of: [tex]A+\lambda v[/tex] but I don't know what to do next.

Thanks!
Find a vector, n1, normal to plane π1, and a vector, n2, normal to plane π2.

How is the line of intersection of the two planes oriented w.r.t. these two normal vectors, n1 and n2 ?
 

What is a line equation?

A line equation is a mathematical representation of a straight line on a graph. It typically takes the form of "y = mx + b" where "m" is the slope of the line and "b" is the y-intercept.

What is a parametric equation?

A parametric equation is a set of equations that express the coordinates of points on a curve or surface in terms of one or more parameters. In the context of a line equation, the parameters are typically "t" or "s" and represent a point along the line.

How do you work out a line equation using two points?

To work out a line equation using two points, you can use the slope formula: m = (y2-y1)/(x2-x1). Once you have the slope, you can plug it into the standard line equation "y = mx + b" along with one of the points to solve for the y-intercept "b".

What is the difference between a line equation and a parametric equation?

The main difference between a line equation and a parametric equation is that a line equation represents a single line on a graph, while a parametric equation represents a curve or surface. In addition, a line equation uses the standard form "y = mx + b" while a parametric equation has a more general form with parameters.

How can line equations and parametric equations be used in real-life applications?

Line equations and parametric equations are used in many real-life applications, such as engineering, physics, and computer graphics. In engineering, they can be used to model the trajectory of a projectile or the path of a moving object. In physics, they can be used to describe the motion of particles in a system. In computer graphics, they are used to create 3D models and animations.

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