Rotational Invariance

Let's take a closer look at why certain vector operations are useful and others are not. Consider the operation of multiplying two vectors component by component to produce a third vector:

g / Component-by-component multiplication of the vectors in 1 would produce different vectors in coordinate systems 2 and 3.

As a simple example, we choose vectors P and Q to have length 1, and make them perpendicular to each other, as shown in figure g/1. If we compute the result of our new vector operation using the coordinate system in g/2, we find:

The x component is zero because Px = 0, the y component is zero because Q_y = 0, and the z component is of course zero because both vectors are in the x - y plane. However, if we carry out the same operations in coordinate system g/3, rotated 45 degrees with respect to the previous one, we find

The operation's result depends on what coordinate system we use, and since the two versions of R have different lengths (one being zero and the other nonzero), they don't just represent the same answer expressed in two different coordinate systems. Such an operation will never be useful in physics, because experiments show physics works the same regardless of which way we orient the laboratory building! The useful vector operations, such as addition and scalar multiplication, are rotationally invariant, i.e., come out the same regardless of the orientation of the coordinate system.