Trajectory as a sufrace intersection

In summary, the given trajectory can be expressed as an intersection of two surfaces, namely the plane z=x and the parabolic cylinder. To find the parametric form for a line given as the intersection of two planes, solve the two linear equations for two variables in terms of the other one and use that as a parameter.
  • #1
Telemachus
835
30

Homework Statement


Well, I must express this trajectory: [tex]\vec{r}=(t^2,2t,t^2)[/tex] as an intersection of two surfaces. I really don't know how to work this. It seems to be some kind of parabola, but I'd really like to see some step by step for solving this.

Bye, and thanks off course.
 
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  • #2
Telemachus said:

Homework Statement


Well, I must express this trajectory: [tex]\vec{r}=(t^2,2t,t^2)[/tex] as an intersection of two surfaces. I really don't know how to work this. It seems to be some kind of parabola, but I'd really like to see some step by step for solving this.

Bye, and thanks off course.

Hint: How are [itex]x(t)[/itex] and [itex]y(t)[/itex] related? How are [itex]z(t)[/itex] and [itex]y(t)[/itex] related?
 
  • #3
Mmm let's see. I got that [tex]x=t^2[/tex], [tex]y=2t[/tex], then [tex](\frac{y}{2})^2=x[/tex]?
And [tex]z=x[/tex]

Is that right? so its the intersection between the plane [tex]z=x[/tex] and the parabolic cylinder...?

I got one more question. I have another exercise where it asks me to get the parametric form for a line given as the intersection of two planes. So, to define the parametric form I need one point that satisfies the equation for the two planes, that is a point over the line. And I find the work of finding such a point really tedious. Is there any technique to find a point that satisfies two equations with three variables? I mean, I'm not much intuitive on that sense, I've been looking for a single point that satisfies the equations for two planes for many minutes and couldn't get it. Is there any procedure to follow?

Bye, and thank you gabba.
 
  • #4
I see no reason to find a specific point if you are looking for the equation of the line itself. If you are given two planes, then you are given two linear equations:
Ax+ By+ Cz= D and Px+ Qy+ Rz= S. You can solve those two equations for two of the variables in terms of the other one. Use that one as parameter.

For example, if the planes are given by 2x- y+ z= 4 and 3x+ y- 2z= 3, then adding the two equations eliminates y and leaves 5x- z= 4. We can solve that for z in terms of x: z= 5x- 4.

Putting that into the first equation, 2x- y+ z= 2x- y+ 5x- 4= 4 so that y= 8- 5x.

Take x as parameter: x= t, y= 8- 5t, z= 5t- 4 are the parametric equations for the line of intersection.
 

Related to Trajectory as a sufrace intersection

1. What is trajectory as a surface intersection?

Trajectory as a surface intersection is a concept used in physics and mathematics to describe the path of an object as it moves through space and intersects with a surface. It is often used to analyze the motion of projectiles or other moving objects.

2. How is trajectory as a surface intersection calculated?

The trajectory of an object can be calculated using mathematical equations and principles such as kinematics, vectors, and calculus. The specific calculations may vary depending on the type of surface and the forces acting on the object.

3. What factors can affect the trajectory of an object?

The trajectory of an object can be affected by several factors, including the initial velocity, angle of launch, air resistance, and gravitational pull. Other factors such as wind, surface friction, and external forces may also have an impact on the trajectory.

4. How is trajectory as a surface intersection used in real life?

Trajectory as a surface intersection is used in various real-life applications, such as in ballistics and sports. It is also used in engineering and design to predict the path of objects and ensure their safe and accurate movement.

5. What are the limitations of using trajectory as a surface intersection?

The accuracy of trajectory as a surface intersection calculations may be limited by various factors, such as environmental conditions, the shape and size of the object, and the complexity of the surface. In addition, unpredictable events or external forces may also affect the trajectory and deviate from the calculated path.

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