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Related Concept Videos

Vector Components in the Cartesian Coordinate System01:29

Vector Components in the Cartesian Coordinate System

Vectors are usually described in terms of their components in a coordinate system. Even in everyday life, we naturally invoke the concept of orthogonal projections in a rectangular coordinate system. For example, if someone gives you directions for a particular location, you will be told to go a few km in a direction like east, west, north, or south, along with the angle in which you are supposed to move. In a rectangular (Cartesian) xy-coordinate system in a plane, a point in a plane is...
Cartesian Vector Notation01:28

Cartesian Vector Notation

Cartesian vector notation is a valuable tool in mechanical engineering for representing vectors in three-dimensional space, performing vector operations such as determining the gradient, divergence, and curl, and expressing physical quantities such as the displacement, velocity, acceleration, and force. By using Cartesian vector notation, engineers can more easily analyze and solve problems in various areas of mechanical engineering, including dynamics, kinematics, and fluid mechanics. This...
Quadratic Models01:23

Quadratic Models

Quadratic models are mathematical representations used to describe relationships in which the rate of change changes at a constant rate. These models appear in a wide variety of natural and engineered systems, especially those involving motion, forces, and optimization. One common application is analyzing the vertical motion of objects influenced by gravity, such as a ball thrown into the air.In such scenarios, the object's height changes over time in a curved pattern, rising to a maximum point...
Vector Algebra: Method of Components01:08

Vector Algebra: Method of Components

It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like addition, subtraction, and multiplication. There are two ways to circumvent this algebraic complexity. One way is to draw the vectors to scale, as in navigation, and read approximate vector lengths and angles (directions) from the graphs. The other way is to use the method of components.
In many applications, the magnitudes and directions of...
Vector Representation of Complex Numbers01:16

Vector Representation of Complex Numbers

Complex numbers, represented in Cartesian coordinates, can also be visualized as vectors. These vectors can be expressed in polar form, emphasizing their magnitude and angle. When a complex number is input into a function, the output is another complex number, highlighting the function's zero point from which the vector representation can originate.
Consider a function defined as the product of the complex factors in the numerator divided by the product of the complex factors in the denominator.
Cartesian Form for Vector Formulation01:26

Cartesian Form for Vector Formulation

The Cartesian form for vector formulation is a process to calculate  the moment of force using the position and force vectors. The moment of force is defined as the cross-product of these vectors, making it a vector quantity. The Cartesian form of the position and force vectors involves unit vectors, which can be used to express the cross-product in determinant form.

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Related Experiment Video

Updated: May 27, 2026

Visualization Method for Proprioceptive Drift on a 2D Plane Using Support Vector Machine
07:05

Visualization Method for Proprioceptive Drift on a 2D Plane Using Support Vector Machine

Published on: October 27, 2016

A note on octonionic support vector regression.

Alistair Shilton, Daniel T H Lai, Braveena K Santhiranayagam

    IEEE Transactions on Systems, Man, and Cybernetics. Part B, Cybernetics : a Publication of the IEEE Systems, Man, and Cybernetics Society
    |November 23, 2011
    PubMed
    Summary
    This summary is machine-generated.

    This study analyzes the octonionic support vector regressor (SVR), revealing limitations in its kernel trick for octonionic data. Applications in gait analysis show its potential compared to other SVR methods.

    Related Experiment Videos

    Last Updated: May 27, 2026

    Visualization Method for Proprioceptive Drift on a 2D Plane Using Support Vector Machine
    07:05

    Visualization Method for Proprioceptive Drift on a 2D Plane Using Support Vector Machine

    Published on: October 27, 2016

    Area of Science:

    • Machine Learning
    • Algebraic Methods
    • Data Analysis

    Background:

    • Introduces the octonionic support vector regressor (SVR) based on Shilton's work.
    • Provides a detailed derivation of the dual form of the octonionic SVR.
    • Identifies three conditions for analogy with the quaternionic SVR case.

    Discussion:

    • Highlights the breakdown of the standard kernel trick with octonionic-valued feature maps.
    • Explores the underlying causes and interpretations of this kernel trick limitation.
    • Proposes methods to extend kernel techniques for octonionic data.

    Key Insights:

    • The octonionic SVR faces challenges with the conventional kernel trick due to its data structure.
    • The study offers insights into overcoming these limitations for broader applicability.
    • Demonstrates the practical application of octonionic SVR in gait analysis.

    Outlook:

    • Investigates potential extensions of kernel methods for octonionic feature maps.
    • Compares the performance of octonionic SVR against least squares, Clifford, and multidimensional SVR.
    • Suggests avenues for future research in non-associative algebraic machine learning.