Vector Product and Its Properties

 

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: