By E. C. Wolstenholme (Auth.)
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50 DIFFERENTIATION OF VECTORS _ dr _ β Then β Ι 1 dr2 . _ + β dt\ _ . η n = /2ii + /3i2 + /fc, Example 2. If r2 = 2/ii — 3/2i2 + 2/3i3, and ^mrf the derivative with respect to t ofx\ X r 2tf«rfofx\. r? Let r — n X Γ2. e. ^ = 3/2(4/3 + 3)ii - 2/(5/ 3 - 2)i2 - 20/3i3 dt Alternatively, the product ri x r2 may be evaluated (see Ch. II, Ex. 2) before being differentiated. Let s = r i . r2. e. i>e< ue · ds A 2 . Ai ^ = (f2ii + /3i2 + ria) · (2ii - 6/i2 + 6^13) at + (2rii + 3/2Ì2 + 13) . (2rii - 3f2i2 + 2««), * = (2f2 - 6t* + 6/3) + (4f2 - 9f4 + 2t% dt * = 6i 2 + 8i 3 - 15/4.
Let a, b, c be represented by OX UB, ÔC respectively. Complete the parallelepiped OBDCAB'D'C, where OA9 BB\ DD\ CC are parallel edges. e. b X c = 2Δηι b x c = am Fig. 23 where ni is the unit vector perpendicular to the plane of b, c such that b, e, DI form a right-handed system. If 0i is the angle between a and m, SCALAR TRIPLE PRODUCT 41 a . (b x c) = a . am = a(a. ni) = a(a cos 0i) = aA = r, where h (=acos0i) is the altitude and V the volume of parallelepiped OBDC AB' D' C", having OBCD as base.
11 that if the vectors a, b are expressed in the forms a = aiii + 02Ì2 + 03Ì3, and b = iiii + £212 + Ì3Ì3, then a . b = (aiii + 0212 + 0313). e. a . b = a\b\ + fl2Ì2 + a*bz- (VII) Also, if b makes an angle Θ with a, a . e. ab cos Θ = a\b\ + αφ% + 0363, cos Θ = a\bl + Ö2&2 + #3^3 ab But a = (al + al + *D* and b = (if + b\ + ig)* and the direction ratios of the lines representing a and b are a\\ a^\ as and i i : Ì2 : Ì3 respectively. e. a x b = (02*3 — 03*2)11 + (03*1 — 01*3)12 + (αι*2 — «2*1)13 (IX) From (IX) it can be seen that the components of the vector a x b are the second order determinants contained in the matrix ( a\ Ü2 a$\ bi b% bzj' Alternatively the result (IX) could be obtained by expanding the determinant 1 ii ia is I a\ Û2 äs (X) b\ bi bz treating ii, 12,13 as if they were real algebraic quantities.
Elementary Vectors by E. C. Wolstenholme (Auth.)