Orthogonal projection onto a plane spanned by two vectors

In summary, the problem involves projecting a point x onto a plane spanned by two vectors, v1 and v2. The projection is found using the projection equation and the cross product between v1 and v2. The final solution is <4.8, 3.6, 0>.
  • #1

Homework Statement


x = <0, 10, 0>
v1 = <4, 3, 0>
v2 = <0, 0, 1>

Project x onto plane spanned by v1 and v2

Homework Equations


Projection equation

The Attempt at a Solution


I took the cross product
k = v1xv2 = <3, -4, 0>

I projected x onto v1xv2
[(x*k)/(k*k)]*k = <-4.8, 6.4, 0 = p

I finished by
x - p = <4.8,3.6,0>

Is this correct?
 
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  • #2
You can check this as follows:

Step 1: check that (x-p) is in the plane, ie check that it can be expressed as a linear combination of v1 and v2.
Step 2: check that (x-p) is perpendicular to p (take the dot product)
Step 3: check that (x-p) + p = p

If these are all correct then the solution is correct.
 

What is orthogonal projection onto a plane spanned by two vectors?

Orthogonal projection onto a plane spanned by two vectors is a mathematical operation that involves finding the closest point on a plane to a given point. It is commonly used in linear algebra and geometry to solve problems involving vectors and planes.

How do you find the orthogonal projection onto a plane spanned by two vectors?

To find the orthogonal projection onto a plane spanned by two vectors, you can use the formula P = (v⋅n/n⋅n)n, where v is the given point, n is a vector that is perpendicular to the plane, and P is the closest point on the plane to v.

What is the purpose of using orthogonal projection onto a plane?

Orthogonal projection onto a plane is used to simplify problems involving vectors and planes. It can also be used to find the shortest distance between a point and a plane, or to determine if a point lies on a plane.

Can orthogonal projection be performed onto a plane spanned by more than two vectors?

Yes, orthogonal projection can be performed onto a plane spanned by any number of vectors. The formula for finding the closest point on the plane will just involve more terms for each additional vector.

Are there real-world applications for orthogonal projection onto a plane?

Yes, there are many real-world applications for orthogonal projection onto a plane. For example, it is used in computer graphics to create 3D images, in engineering to find the optimal angle for a ramp or roof, and in physics to calculate the trajectory of a projectile on a flat surface.

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