By E. Wolstenholme, W. J. Langford and E. A. Maxwell (Auth.)
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Extra resources for Elementary Vectors
S). 1 Suppose the vector r is a continuous single-valued function of the scalar variable L Suppose that when t increases by a small scalar quantity 8/, r increases by a small vector quantity 8r. Then the derivative of r with respect to t is defined to be —= dt hm —. 8t Thus the existence of the derivative depends upon the existence of the limit hm ~ The derivative of r, when it exists, is in general also a function of t, and if dr/dt is differentiable its derivative is the second derivative of Γ with respect to / and is written d^r/dtK Similarly for the third, fourth, .
R + ... etc. 8 By definition, if the vector b makes an angle Θ with the vector a, and η is a real number n(a . b) = nab cos θ also (n&). b = nab cos Θ and a . (nh) = nab cos Θ, Hence n(a . b) = («a). b = a . 9· Distributiye Law—Vector Product (i) Suppose the vectors a and b are represented by OA and 0 5 and suppose Π is the plane through Ο perpendicular to OA. Let BB' be the perpendicular from 5 to Π and let b' be the vector represented by 0B\ Then if b makes an angle θ mth a, b'=^b sin θ Fig. 20 Suppose ni is the unit vector perpendicular to a, b so that a, b, ni form a right-handed system, then a X b= sin θ)ηι.
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 + . . ) . ( p + q + r + . . q + A . r + . . ) = p . a + p . b + p . c + ...