Simple Nature - Light and Matter

A useless vector operation example 58The way I’ve defined the various vector operations above aren’tas arbitrary as they seem. There are many different vector operationsthat we could define, but only some of the possible definitionsare mathematically useful. Consider the operation of multiplyingtwo vectors component by component to produce a thirdvector:R x = P x Q xR y = P y Q yR z = P z Q zAs a simple example, we choose vectors P and Q to have length1, and make them perpendicular to each other, as shown in figurek/1. If we compute the result of our new vector operation usingthe coordinate system shown in k/2, we find:k / Two vectors, 1, to whichwe apply the same operation intwo different frames of reference,2 and 3.R x = 0R y = 0R z = 0The x component is zero because P x = 0, the y component iszero because Q y = 0, and the z component is of course zero becauseboth vectors are in the x-y plane. However, if we carry outthe same operations in coordinate system k/3, rotated 45 degreeswith respect to the previous one, we findR x = −1/2R y = 1/2R z = 0The operation’s result depends on what coordinate system weuse, and since the two versions of R have different lengths (onebeing zero and the other nonzero), they don’t just represent thesame answer expressed in two different coordinate systems. Suchan operation will never be useful in physics, because experimentsshow physics works the same regardless of which way we orientthe laboratory building! The useful vector operations, such asaddition and scalar multiplication, are rotationally invariant, i.e.,come out the same regardless of the orientation of the coordinatesystem.All the vector techniques can be applied to any kind of vector,but the graphical representation of vectors as arrows is particularlynatural for vectors that represent lengths and distances. We define avector called r whose components are the coordinates of a particularpoint in space, x, y, and z. The ∆r vector, whose components are∆x, ∆y, and ∆z, can then be used to represent motion that starts at196 Chapter 3 Conservation of Momentum