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Vector Product and Its Properties

Vector Product and Its Properties - раздел Образование, Chapter 2. Analytic Geometry   We Have Considered A Product Of Two Vectors Equal To A Number...

 

We have considered a product of two vectors equal to a number (inner product)

.

What happens if the product of two vectors is a vector:

?

Consider two vectorsand :

 

Definition.The vector product of two vectors and is a vector , satisfying the following conditions:

(1) the absolute value of equals the product of the absolute values of the two given vectors and the sine of the angle between them:

; (*)

(2) the vector is perpendicular to both vectors and :

;

(3) the three vectors ,and constitute a right triple of vectors (that is, looking from the tail of, we see that the shorter rotation from to is carried out anticlockwise). The vector product of and is denoted by

.

Properties of vector product.

1.The absolute value of the vector product of two vectors is equal to the area of the parallelogram spanned by these vectors:

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Эта тема принадлежит разделу:

Chapter 2. Analytic Geometry

Vector Algebra Operations on Vectors... Definition A directed interval or an ordered pair of points is called a vector...

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Все темы данного раздела:

Linear Operations on Vectors
Definition. The product of a vector and a real number l is the vector

Decomposition of vectors.
Theorem 1. An arbitrary vectorin the plane can be decomposed into two noncollinear vectors:

When a vector is multiplied by a number l, each coordinate of this vector is multiplied by this number.
Remark.Relation (2) is a written in vector notation; the coordinate notation of a vector is

Inner Product of Vectors and its Properties
Definition. The inner product of two vectors and

The inner product of vectors in coordinates. Consider two vectors
and . The last proper

The inner product of vectors is equal to the sum of products of their coordinates.
Example 1. -? and

The direction of a vector. Let us find the angle between two vectorsand .
Consider the inner product . We have

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