Dot operations are straightforward when each dimension only operates with values in its own dimension, but the linearized expressions are easier to remember for the students than the vector space orientation pictures. The four basic dot operations on two vectors are shown in Fig. . When the unit vector arrows that are parallel to the axis are in the same direction as the axis vector, the value is positive, and if in the opposite direction, the operations are negative (Fig. ). Dot products are straightforward in that each dimension is independent of the others and only cares about calculations of values in the same dimension (Fig. ). One interesting thing is that when the two vectors are perpendicular to each other the dot product vector (Fig. ) will be oriented at -135 degrees which has the same amplitude on both axes but with opposite signs.