The cross product is a fundamental concept in vector algebra that is a vector operation on two different vectors to obtain a third vector. Unlike the scalar product, the cross product results in a vector quantity perpendicular to both the original vectors.

The magnitude of the cross product is obtained by multiplying the magnitude of both the vectors and the sine of the angle between them. This means that a larger angle between the vectors will lead to a greater magnitude of the cross product.

The direction of the resultant vector is determined by using the right-hand rule. If we curl the fingers of the right hand from vector A to vector B, then the direction of the thumb represents the direction of the resultant vector. In other words, the direction of the cross product is perpendicular to the plane containing vectors A and B.

It is important to note that the cross product of vectors is non-commutative. That is, the resultant vector for the cross product of vector A with vector B is equal in magnitude but opposite in direction to that of the cross product of vector B with vector A. However, the cross product obeys the associative law and distributive laws of addition. This means that the cross product of a sum of vectors equals the sum of the cross products of each vector.

In conclusion, the cross product is a fundamental concept in vector algebra that has widespread applications in physics and engineering, such as calculating the moment of a force about a point, calculating the torque of a force about an axis, and calculating the angular momentum of a body about an axis.