Vector Cross Product : First Principles
» multiplied with component in perpendicular
→ product of magnitudes of components in perpendicular
→ direction perpendicular to both the vectors
The product where vector components in perpendicular interact is defined. Given vectors and as shown in figure. The component of perpendicular to the us
The angle between the vectors is . Then
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.
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 .
The direction of the cross product 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 to , defines the normal. s
Another form to find the direction of the cross product 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 to , defines the normal.
Formal definition of vector cross product.
where is the normal denoting the direction of right handed rotation from to .
Vector Cross Product: for vectors
where is the unit vector of right-handed rotation from to .
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
→ 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