Proving Orthogonal Projection of Triangle V, v'_{1}

In summary, orthogonal projection is a mathematical operation used to project a vector or point onto a subspace in a direction perpendicular to that subspace. It is commonly used in geometry to preserve distances and angles between objects and their projections. To prove that a triangle is orthogonally projected onto a vector, one can show that the projection of each vertex is perpendicular to the vector. This is important for accurately representing three-dimensional objects on a two-dimensional plane and has practical applications in various fields. Additionally, a triangle can be orthogonally projected onto multiple vectors, as seen in real-world scenarios.
  • #1
rannasquaer
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Given the triangle above where \(\displaystyle V < v'_{1}\), prove that the \[ v_{1}=V \cos(\psi)+v'_{1} \cos(\theta - \psi) \]

It is said that \(\displaystyle v_{1}\) is equal to the sum of the orthogonal projections on \(\displaystyle v_{1}\) of \(\displaystyle V\) and of \(\displaystyle v'_{1}\) and that is precisely the expression that show. But I couldn't see how to make the projection and the calculations.
 
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  • #2
1607353732937.png
 

FAQ: Proving Orthogonal Projection of Triangle V, v'_{1}

What is orthogonal projection?

Orthogonal projection is a mathematical concept that involves projecting a vector or shape onto a plane or subspace in a way that preserves the right angles between the original vector and the projected vector.

What is a triangle V, v'_{1}?

A triangle V, v'_{1} is a triangle with three vertices, V, and a vector v'_{1} that is perpendicular to one of the sides of the triangle.

How do you prove orthogonal projection of a triangle V, v'_{1}?

To prove orthogonal projection of a triangle V, v'_{1}, you can use the Pythagorean theorem and the dot product to show that the projected vector is perpendicular to the original vector and that the length of the projected vector is equal to the length of the original vector multiplied by the cosine of the angle between them.

What are some real-world applications of orthogonal projection?

Orthogonal projection has many applications in fields such as engineering, computer graphics, and physics. It is used to create 3D models, map objects onto surfaces, and calculate forces and velocities in mechanics.

How does orthogonal projection differ from other types of projections?

Orthogonal projection is unique in that it preserves the angles between the original vector and the projected vector, whereas other types of projections may distort or change these angles. It is also a special case of a more general concept called vector projection.

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