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Cross Product: First Principles

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Vector Cross Product : First Principles

 »  multiplied with component in perpendicular
    →  p×q =p×b =|p||b|n^

    →  product of magnitudes of components in perpendicular

    →  direction perpendicular to both the vectors

in perpendicular

vector cross product first principles

The product where vector components in perpendicular interact is defined. Given vectors p and q as shown in figure. The component of q perpendicular to the p us b

The angle between the vectors is θ. Then |b| =|q|sinθ

vector cross product first principles

What will be a good choice of direction of the product between components in perpendicular?

Two intersecting lines, which are not parallel, define a plane. The normal on the face of the plane describes the plane and so, the normal is taken as the direction of the product.

vector cross product choice of directions

A plane can be described by

 •  normal which is one side of the plane, or

 •  the negative of the normal, which is the other side of the plane.

One of this can be chosen as the direction of cross product p×q.

vector cross product standard screw direction

The direction of the cross product p×q is shown in the figure. In a standard right-handed rectangular coordinate system,
The direction in which a standard screw advances when it turns from p to q, defines the normal. s

vector cross product right hand thumb rule for direction

Another form to find the direction of the cross product p×q is shown in the figure. In a standard right-handed rectangular coordinate system,
The direction pointed by the right thumb when fingers are curled to point from p to q, defines the normal.

vector cross product illustration

Formal definition of vector cross product.
p×q=|p||q|sinθn^ where n^ is the normal denoting the direction of right handed rotation from p to q.


Vector Cross Product: for vectors p,qR3
where n^ is the unit vector of right-handed rotation from p to q.

Vector Cross Product is defined as the product of components in perpendicular.


The outline of material to learn vector-algebra is as follows.

Note: Click here for detailed outline of vector-algebra.

•   Introduction to Vectors

    →   Introducing Vectors

    →   Representation of Vectors

•   Basic Properties of Vectors

    →   Magnitude of Vectors

    →   Types of Vectors

    →   Properties of Magnitude

•   Vectors & Coordinate Geometry

    →   Vectors & Coordinate Geometry

    →   Position Vector of a point

    →   Directional Cosine

•   Role of Direction in Vector Arithmetics

    →   Vector Arithmetics

    →   Understanding Direction of Vectors

•   Vector Addition

    →   Vector Additin : First Principles

    →   Vector Addition : Component Form

    →   Triangular Law

    →   Parallelogram Law

•   Multiplication of Vector by Scalar

    →   Scalar Multiplication

    →   Standard Unit Vectors

    →   Vector as Sum of Vectors

    →   Vector Component Form

•   Vector Dot Product

    →   Introduction to Vector Multiplication

    →   Cause-Effect-Relation

    →   Dot Product : First Principles

    →   Dot Product : Projection Form

    →   Dot Product : Component Form

    →   Dot Product With Direction

•   Vector Cross Product

    →   Vector Multiplication : Cross Product

    →   Cross Product : First Principles

    →   Cross Product : Area of Parallelogram

    →   Cross Product : Component Form

    →   Cross Product : Direction Removed