Find an expression for the orthogonal projection of a point v onto a plane P that is characterized by θ and θ0. Write your answer in terms of v, θ and θ0.
(Enter theta_0 for the offset θ0.
Enter norm(theta) for the norm ∥θ∥ of a vector θ.
Use * to denote the dot product of two vectors, e.g. enter v*w for the dot product v⋅w of the vectors v and w. )
To find the expression for the orthogonal projection of a point v onto a plane P characterized by θ and θ0, we can use the formula:
proj(v) = v - ((v∙θ - θ0) / ∥θ∥²) * θ
where:
- v is the point we want to project onto the plane,
- θ is the normal vector of the plane,
- θ0 is the offset from the origin.
Let's substitute the variables in:
proj(v) = v - ((v∙θ - θ0) / ∥θ∥²) * θ
Now we have the expression for the orthogonal projection of a point v onto a plane P characterized by θ and θ0.
The orthogonal projection of a point v onto a plane P can be found using the formula:
projP(v) = v - ((v * n) / ∥n∥^2) * n
Where v is the point being projected, n is the normal vector of the plane P, and * denotes the dot product.
In this case, we are given that the plane P is characterized by θ and θ0. To find the normal vector n, we can use the formula:
n = (cos(θ) * cos(θ0), sin(θ) * cos(θ0), sin(θ0))
Thus, the expression for the orthogonal projection of a point v onto the plane P characterized by θ and θ0 is:
projP(v) = v - ((v * n) / ∥n∥^2) * n
Substituting the expression for n, we get:
projP(v) = v - ((v * (cos(θ) * cos(θ0), sin(θ) * cos(θ0), sin(θ0))) / (∥(cos(θ) * cos(θ0), sin(θ) * cos(θ0), sin(θ0))∥^2)) * (cos(θ) * cos(θ0), sin(θ) * cos(θ0), sin(θ0))