Answer to Question #265700 in Analytic Geometry for Almas

Question

Definition, Formula and Physical interpretation of Cross Product.

Answer 1

The cross product of two vectors is the third vector that is perpendicular to the two original vectors. Its magnitude is given by the area of the parallelogram between them and its direction can be determined by the right-hand thumb rule.

cross product is defined by the formula:

$a\times b=\overline{n}||a||||b||sin\theta$

where θ is the angle between a and b in the plane containing them,

‖a‖ and ‖b‖ are the magnitudes of vectors a and b,

n is a unit vector perpendicular to the plane containing a and b, in the direction given by the right-hand rule

Matrix notation:

$a\times b=\begin{vmatrix} i & j &k\\ a_1 & a_2&a_3\\ b_1 & b_2&b_3 \end{vmatrix}$

Physical interpretation:

Cross products are a kind of measure of "difference" between two vectors. With a cross product the more perpendicular your two vectors are the higher your cross product's magnitude will be.