1. Cross product of Two vectors
Definition
The cross product of two vectors is two dimensional concept.
It is a vector expressing the angular relationship between the
vectors.
It is a vactor value as an operation of two vectors with
the same number of components (at least three).
Let's say, we have two vectors, 𝑎 and 𝑏, if |𝑎| and |𝑏|
represent the lengths of vectors 𝑎 and 𝑏, respectively, and
if 𝜃 is the angle between these vectors.
Then, The cross product of vectors 𝑎 and 𝑏 will have the following
relationship:
𝑎 × 𝑏 = |𝑎||𝑏|sin 𝜃

2. Cross product of Two vectors
Geometrical Interpretation
Given the characteristics of the cross product of two vectors by the
relation
𝑎 × 𝑏 = |𝑎||𝑏|sin 𝜃
Now, we can interpret three possible conditions:
1. 𝑎 × 𝑏 is perpendicular to both the vectors 𝑎 and 𝑏.
2. 𝑎 × 𝑏 represents the area of parallelogram determined by
the these vectors as adjacent sides.
3. If 𝑎 and 𝑏 are parallel vectors then 𝑎 × 𝑏 = 0

3. Let 𝒂 and 𝒃 be vectors and consider the parallelogram that the two vectors make.
Then
||𝒂 × 𝒃|| = Area of the Parallelogram
and the direction of 𝒂 × 𝒃 is a right angle to the parallelogram that follows the right
hand rule
Note:
For 𝒊 × 𝒋 the magnitude is 1 and the direction is 𝒌, hence 𝒊 × 𝒋= 𝒌.

4. More generally,
The magnitude of the product equals the area
of a parallelogram with the vectors for sides.
In particular for perpendicular vectors this is a
rectangle and the magnitude of the product is
the product of their lengths.
The cross product
is anticommutative, distributive over addition

5. The cross product
(vertical -green)
changes
as the angle
between the vectors
(black and red)
changes