Vectors and rotors, with applications

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1903 Excerpt: ...the division therefore belonging to this multiplication is not a unique operation. The geometrical meaning of the theorem is: Parallelograms on the same base and between the same parallels are equal. If the area and base are given, the other side may be any vector drawn between the parallels. The student should be careful to notice that from the equation aft = ay it does not follow that /3 = 7, but, supposing that a == 0, ft = y + ma where m is any scalar. The equation can also be written aft-y = 0, which as we have seen is only true if either a = 0, or ft = y, ovft--ya. The latter says ft--y--ma as above. If however a/3 = ay shall be true whatever a may be, then of necessity ft' = y, because ft--y cannot be parallel to two different directions. The two equations a0 = 07, a'ft = a'y, where a and a are not parallel, can only be true if ft = y. Such reasoning enables us to dispense with the operation of Division by Vectors. This operation is complicated and will not be considered at all. It leads to the much more complicated Theory of Quaternions. 130. The remainder of this chapter will be devoted to various direct applications of the preceding theory to Pure and to Coordinate Geometry, &c. A number of illustrative examples will be worked in order to shew the manner in which problems should be attacked vectorially. The chapters on Rotors and Stress diagrams may however be taken before reading the remainder of the chapter. 131. Let a and /3 denote the adjacent sides of any parallelogram as vectors, then the diagonals are a+/3 and o-/3. Now a+01 a-/3=aa-a/3 + /3a-0/3 = 2 fia. (Notice that unless the proper order of the vectors had been retained in the multiplication, the third term might have cancelled with the second.) This result says:--the area of any pa...