Multiplication of Vectors by a Real Number or Scalar

The scalar or dot product of two vectors \[\vec A\] and \[\vec B\] defined as the product of the magnitudes of \[\vec A\] and \[\vec B\] and cosine of the angle θ between them is called the scalar product or dot product.

The vector or cross product of two vectors is defined as the vector whose magnitude is equal to the product of the magnitudes of two vectors and sine of the angle between them, and whose direction is perpendicular to the plane of the two vectors and is given by the right hand rule — this is called the vector product or cross product.

The rule which states that if we curl the fingers of the right hand in such a way that they point in the direction of rotation from vector \[\vec A\] to \[\vec B\] through the smaller angle, then the stretched thumb points in the direction of \[\vec A\] × \[\vec B\] is called the Right Hand Thumb Rule.