Find parametric equations given point and two planes

In summary, to find the parametric equations parallel to the line of intersection between 2x-y+z=1 and 6x-y-z=3 and passing through the point (5,-1,3), we first find the normal vectors for both planes, which are <2,-1,1> and <6,-1,-1>. Taking the cross product of these normal vectors, we get the vector <2,-4,4>. This vector represents the direction of the parametric equations, which can be written as x=5+2t, y=-1-4t, z=3+4t, where 0≤t≤1.
  • #1
getty102
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Homework Statement



Find the parametric equations through point (5,-1,3) parallel to the line of intersection between 2x-y+z=1 and 6x-y-z=3, where 0≤t≤1

Homework Equations


1. Find normal vectors for both planes
2. Take cross product of both normal planes
...


The Attempt at a Solution


normal to plane 1 is <2,-1,1>
normal to plane 2 is <6,-1,-1>

cross product to both normal vectors is (1+1)i-(-2+6)j+(-2+6)k
=2i-4j+4k=<2,-4,4>

I wasn't sure how to input the point or if I am on the right track?
 
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  • #2
You're on the right track. What do the parametric equations for a line generally look like?
 

FAQ: Find parametric equations given point and two planes

What are parametric equations?

Parametric equations are a set of equations that express the coordinates of a point in terms of one or more independent variables, called parameters. These equations are often used to describe curves, surfaces, and other mathematical objects.

How do I find parametric equations for a point and two planes?

To find parametric equations for a point and two planes, you can use the method of intersecting lines. This involves finding the intersection point of the two planes and then using that point as the starting point for a line that lies on both planes. The direction of the line can be determined by finding the cross product of the normal vectors of the two planes.

What information do I need to find parametric equations?

To find parametric equations, you will need the coordinates of a point on the desired curve or surface, as well as the equations of two planes that the curve or surface intersects. Additionally, you will need to know the direction of the curve or surface, which can be determined by finding the cross product of the normal vectors of the two planes.

Can I use parametric equations to describe any curve or surface?

Yes, parametric equations can be used to describe a wide variety of curves and surfaces, including lines, circles, ellipses, parabolas, hyperbolas, and more complex curves and surfaces. However, in some cases, other methods may be more efficient or accurate.

What are some real-world applications of parametric equations?

Parametric equations have numerous applications in fields such as physics, engineering, and computer graphics. They are commonly used to describe the motion of objects, such as projectiles or planets, as well as the shapes of objects, such as bridges or roller coasters. They are also used in computer graphics to create smooth and realistic curves and surfaces.

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