By recalling how the dot and cross products between pairs of the three mutually perpendicular unit vectors along x, y and z are given - you can explicitly find the dot and cross products by explicit multiplication.
Vectors - Rotating the Coordinate system
The specific components of a vector depend on the coordinate system, but the length of the vector remains unchanged as it should. Shown below are the equations that relate components in one coordinate system to the components in another coordinate system where one coordinate system is rotated with respect to the other.
