Orthogonal Decomposition Calculator

Orthogonal Decomposition Formula:

\[ \vec{v} = \text{proj}_{\vec{w}} \vec{v} + (\vec{v} - \text{proj}_{\vec{w}} \vec{v}) \]

Vector v (x-component):

Vector v (y-component):

Vector v (z-component):

Vector w (x-component):

Vector w (y-component):

Vector w (z-component):

Projection of v onto w (proj_w v):

Orthogonal Component (v - proj_w v):

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1. What is Orthogonal Decomposition?

Orthogonal decomposition breaks a vector into two components: one parallel to a given vector and one perpendicular to it. This is fundamental in vector analysis and has applications in physics, engineering, and computer graphics.

2. How Does the Calculator Work?

The calculator uses the orthogonal decomposition formula:

\[ \vec{v} = \text{proj}_{\vec{w}} \vec{v} + (\vec{v} - \text{proj}_{\vec{w}} \vec{v}) \]

Where:

\( \vec{v} \) — The original vector to be decomposed
\( \vec{w} \) — The vector we're projecting onto
\( \text{proj}_{\vec{w}} \vec{v} \) — The projection of v onto w (parallel component)
\( \vec{v} - \text{proj}_{\vec{w}} \vec{v} \) — The orthogonal component

Explanation: The projection is calculated using the formula \( \text{proj}_{\vec{w}} \vec{v} = \left( \frac{\vec{v} \cdot \vec{w}}{\vec{w} \cdot \vec{w}} \right) \vec{w} \).

3. Importance of Orthogonal Decomposition

Details: Orthogonal decomposition is crucial in physics for separating forces into components, in computer graphics for shading calculations, and in machine learning for dimensionality reduction.

4. Using the Calculator

Tips: Enter the components of vectors v and w. The calculator will compute both the projection of v onto w and the remaining orthogonal component.

5. Frequently Asked Questions (FAQ)

Q1: What if vector w is the zero vector?
A: The projection is undefined since you can't project onto a zero vector. The calculator will show an error.

Q2: Can I use this for 2D vectors?
A: Yes, just set the z-components to 0 for both vectors.

Q3: How is this different from vector resolution?
A: Orthogonal decomposition specifically creates components that are perpendicular to each other, while general vector resolution can use any basis.

Q4: What applications does this have in physics?
A: It's used to separate forces into components parallel and perpendicular to surfaces, analyze motion on inclined planes, and more.

Q5: Can this be extended to higher dimensions?
A: Yes, the same principle applies in any finite-dimensional vector space.