The calculator is designed to find a vector that is orthogonal, or perpendicular, to another given vector. This capability is essential in various applications, including computer graphics, where orthogonal vectors define normal surfaces to render scenes correctly.
Formula of Find Orthogonal Vector Calculator
To determine an orthogonal vector to a given vector v = [v1, v2, v3], we employ the equation v dot u = 0. Here, u = [u1, u2, u3] represents the orthogonal vector. The dot product v1u1 + v2u2 + v3u3 = 0 must be zero, which means the vectors are orthogonal. By manipulating this formula, one can derive multiple possible orthogonal vectors, providing flexibility in application.
Tools and Resources
| Input Vector (v) | Orthogonal Vector (u) | Notes |
|---|---|---|
| [1, 0, 0] | [0, 1, 0] | u is perpendicular to v along the y-axis |
| [0, 1, 0] | [0, 0, 1] | u is perpendicular to v along the z-axis |
| [0, 0, 1] | [1, 0, 0] | u is perpendicular to v along the x-axis |
| [1, 1, 0] | [0, 0, 1] | u is perpendicular to the xy-plane |
| [1, 0, 1] | [0, 1, 0] | u lies on the y-axis, perpendicular to v |
| [0, 1, 1] | [1, 0, 0] | u lies on the x-axis, perpendicular to v |
| [1, 1, 1] | [-1, 1, 0] | u is one possible orthogonal vector |
| [2, 3, 5] | [-15, 10, 0] | Example detailed in the blog post |
| [3, 4, 0] | [0, 0, 1] | Orthogonal in 2D extended into 3D with z-component as 1 |
| [1, 2, 3] | [-2, 1, 0] | Solving for an orthogonal vector in 3D space |
Example of Find Orthogonal Vector Calculator
Consider a vector v = [2, 3, 5]. To find a vector orthogonal to v, input these components into the calculator. The tool computes potential orthogonal vectors such as u = [-15, 10, 0], among others, demonstrating the practical application of the underlying formula.
Most Common FAQs
An orthogonal vector is one that forms a right angle with another vector. This property is crucial in various mathematical and physical contexts.
The calculator can handle both 2D and 3D vectors. Users simply input the components corresponding to their vector's dimensionality.
Finding orthogonal vectors is essential in fields like robotics (for motion planning) and computer graphics (for lighting and shading models).