By E. Wolstenholme, W. J. Langford and E. A. Maxwell (Auth.)
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Extra info for Elementary Vectors
Thus the distributive law is obeyed in the scalar multiplication of vectors. 7 ii) can be extended to include a product of the form (a + b + c + . . ) . ( p + q + r + . . ) . For suppose that a + b + c . . = A, then (a + b + c + . . ) . ( p + q + r + . . q + A . r + . . ) = p . a + p . b + p . c + ... = a . p ^ b . p + ... Similarly A . q - a . q + b . q + c . q + . . , etc. Hence (a + b + c + . . ) . ( p + q + r + . . ) = a . p + b . p + c . p + ... q + c . q + . . + a . r + ... etc.
7, resolving in the direction of a, Γ therefore cos ^ = 6 cos Ö + c cos φ a . r = a ( r cos φ) = αφ cos θ + c cos Φ) = a . b + a . e. a . (b + c) = a . b + a . c. This argiunent can be extended to include any nxunber of vectors, so that a . ) = a . b + a . c + a . d + ... Thus the distributive law is obeyed in the scalar multiplication of vectors. 7 ii) can be extended to include a product of the form (a + b + c + . . ) . ( p + q + r + . . ) . For suppose that a + b + c . . = A, then (a + b + c + .
Thus the vector product of two vectors does not obey the com mutative law. Adopting this definition of a vector product the moment of the force F about the origin is s χ F where s is the position vector of the point of application of F. 7. c = cos Ö) + a(c cos φ) = αφ cos ö + c cos Φ) -C Fig. 19 If r is the smn of b and c making an angle φ with a, then a . Γ =fl/*cos φ == a(r cos Φ) But r = b + c. 7, resolving in the direction of a, Γ therefore cos ^ = 6 cos Ö + c cos φ a . r = a ( r cos φ) = αφ cos θ + c cos Φ) = a .
Elementary Vectors by E. Wolstenholme, W. J. Langford and E. A. Maxwell (Auth.)