Projection and Reflection of Vector WRT plane

In summary: Good job! In summary, the conversation discusses calculating the projection and reflection of a given vector onto a plane with a given normal vector. The correct solutions involve normalizing the normal vector and using the formula v - (v \cdot \hat{n})\hat{n} for the projection, and v - 2(v \cdot \hat{n})\hat{n} for the reflection. The final solutions are Pv=\frac{1}{6}(25i+10j-5k) and Tv=\frac{1}{6}(32i-4j+2k).
  • #1
SP90
23
0

Homework Statement



Given a plane [itex]\Pi[/itex] with normal [itex]n=i-2j+k[/itex] and a vector [itex]v=3i+4j-2k[/itex] calculate the projection of [itex]v[/itex] onto [itex]\Pi[/itex] and the reflection of [itex]v[/itex] with respect to [itex]\Pi[/itex].

The Attempt at a Solution



I need to check that I'm doing this is right.

I think I need [itex]v - (v \cdot n)n = 3i+4j-2k - 7(i-2j+k) = -4i +18j-9k[/itex]

And for the refection:

[itex]v - 2(v \cdot n)n = 3i+4j-2k - 14(i-2j+k) = -11i +32j-16k[/itex]

Are those correct?
 
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  • #2
SP90 said:

Homework Statement



Given a plane [itex]\Pi[/itex] with normal [itex]n=i-2j+k[/itex] and a vector [itex]v=3i+4j-2k[/itex] calculate the projection of [itex]v[/itex] onto [itex]\Pi[/itex] and the reflection of [itex]v[/itex] with respect to [itex]\Pi[/itex].

The Attempt at a Solution



I need to check that I'm doing this is right.

I think I need [itex]v - (v \cdot n)n = 3i+4j-2k - 7(i-2j+k) = -4i +18j-9k[/itex]
You made a sign error.

The projection of v onto the plane should be perpendicular to the plane, right? So what should your answer dotted with ##\vec{n}## equal? That's how you can check your answer.
SP90 said:
And for the refection:

[itex]v - 2(v \cdot n)n = 3i+4j-2k - 14(i-2j+k) = -11i +32j-16k[/itex]

Are those correct?
Same sign error.
 
  • #3
Makes sense, thanks for the quick response.

I get [itex]10i-10k+5k[/itex] which, when dotted with [itex]n[/itex] obviously gives 0.

Correcting the sign error for the second yields [itex]17i-24j+12k[/itex].

Thanks for your help
 
  • #4
That's not correct either. You need to normalize the normal vector before you use it in your formulas.
 
  • #5
Ah, I see.

So instead I use [itex]\hat{n}=\frac{1}{\sqrt{6}}(1i-2j+1k)[/itex]

and that gives [itex]Pv=\frac{1}{6}(25i+10j-5k)[/itex].

And [itex]Tv=\frac{1}{6}(32i-4j+2k)[/itex].

Is that right now?
 
  • #6
Yes, those are correct.
 

1. What is the difference between projection and reflection of a vector with respect to a plane?

Projection of a vector onto a plane is the perpendicular distance from the vector to the plane, while reflection of a vector with respect to a plane is the mirror image of the vector across the plane.

2. How do you calculate the projection of a vector onto a plane?

The projection of a vector onto a plane can be calculated by finding the dot product between the vector and the normal vector of the plane, and then dividing it by the magnitude of the normal vector. This gives the length of the projection vector.

3. Can the projection of a vector onto a plane be negative?

Yes, the projection of a vector onto a plane can be negative if the vector and the normal vector of the plane are in opposite directions.

4. What is the formula for reflecting a vector with respect to a plane?

The formula for reflecting a vector with respect to a plane involves finding the perpendicular distance from the vector to the plane, and then adding twice this distance to the original vector in the opposite direction of the normal vector of the plane.

5. How is the projection and reflection of a vector with respect to a plane useful in real-life applications?

The projection and reflection of a vector with respect to a plane are useful in many areas such as computer graphics, engineering, physics, and navigation. They are used to create 3D models, design structures, calculate forces and velocities, and determine the trajectory of objects in space.

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