The Art and Craft of Problem Solving

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8.5 TRANSFORMATIONS 299

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To indicate the motion along a side of triangle ABC, we need vector subtraction. The
vector OC denotes the (relative) motion from B to C. This can also be written as
OC=C-B.
Remember that these are all relative motions. We can interpret C - B to mean, "travel
from the origin to C, then perform the vectorial motion of B, only going backwards ."

We start at 0, travel to C, then end up at P. Notice that OP and BC are parallel

segments with equal length. Thus the relative motion "start at 0 , end up at P" is equal

to the relative motion "start at B, end up at C." As vectors, the two are absolutely the
same: oP = OC.
The shaded triangles in the figure above (on the right) illustrate translation by A.
If we denote this translation by T, we have

T(f::::.EFG) = f::::.E'F'G',

since

£' -£ =F'-F =8'-8 =A.
Now let's recall vector addition. Undoubtedly, you recall the "parallelogram rule."
But first, think of vector addition merely as composition of motions. In other words,
the vector sum X + Y means, "do the (relative) motion X, followed by the motion Y.
For example, in the figure above, it should be clear that oP + B = C. One way to see
this is with the parallelogram rule (OC is the diagonal of the parellelogram with sides
OP and OB), and another way is compose motions: First perform oP, which is the

(relative) motion from the origin to P; then perform B. SO if we started at 0, we end

at C. (Of course, if we started somewhere else, we'd end up somewhere else, but the
motion would be parallel and equal in length to the motion from from 0 to C.
Here are two simple and well known consequences of these ideas:

• The vector sum along the sides of a closed polygon is zero. For example, in the
  figure above, we have