An Elementary Treatise on Quaternions |
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Page 7
... respectively to any three given vectors , no two of which are parallel , and which are not parallel to one plane . Let OA , OB , OC be the three fixed vectors , c OP any other vector . From P draw PQ parallel to CO , meeting the plane ...
... respectively to any three given vectors , no two of which are parallel , and which are not parallel to one plane . Let OA , OB , OC be the three fixed vectors , c OP any other vector . From P draw PQ parallel to CO , meeting the plane ...
Page 8
... respectively , in these three expressions , AB + BC + CD = AB + ( BC + CD ) = ( AB + BC ) + CD . And thus the truth of the associative law is evident . 28. ] The equation p = x B , where p is the vector connecting a variable point with ...
... respectively , in these three expressions , AB + BC + CD = AB + ( BC + CD ) = ( AB + BC ) + CD . And thus the truth of the associative law is evident . 28. ] The equation p = x B , where p is the vector connecting a variable point with ...
Page 12
... respectively ; the locus of Q is a straight line parallel M to OA . B P ช At Q we have , therefore , xy = 1+ zx , y ( e − 1 ) = ze . = } Let OM = ea . Then AP e - 1a + xß . Hence the equation of OQ is p = y ( e - la + xß ) ; and that ...
... respectively ; the locus of Q is a straight line parallel M to OA . B P ช At Q we have , therefore , xy = 1+ zx , y ( e − 1 ) = ze . = } Let OM = ea . Then AP e - 1a + xß . Hence the equation of OQ is p = y ( e - la + xß ) ; and that ...
Page 14
... respectively the coefficients of a and ß . Hence t = p ± √ p2 -2q . Thus , in general , two tangents can be drawn from a given point . These coincide if p2 = 2q ; that is , if the vector of the point from which they are to be drawn is ...
... respectively the coefficients of a and ß . Hence t = p ± √ p2 -2q . Thus , in general , two tangents can be drawn from a given point . These coincide if p2 = 2q ; that is , if the vector of the point from which they are to be drawn is ...
Page 15
... respectively , it is obvious that the vector join- ing the points of contact is 2 al2 + Bl2 2 Bt2 - - at2— 2 which is parallel to a + ß tz a + plitte ; 2 or , by the values of t1 and t2 in ( g ) , a + pß . Its direction , therefore ...
... respectively , it is obvious that the vector join- ing the points of contact is 2 al2 + Bl2 2 Bt2 - - at2— 2 which is parallel to a + ß tz a + plitte ; 2 or , by the values of t1 and t2 in ( g ) , a + pß . Its direction , therefore ...
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Common terms and phrases
a₁ axes axis Cartesian centre of inertia Chapter circle cloth cone conjugate constant cöordinates coplanar curvature curve developable surface diameters differential direction drawn easily ellipsoid envelop equal evidently expression Extra fcap extremity fcap Find the equation Find the locus given equation given line given vectors gives Hamilton Hence hodograph integral intersection last section length linear and vector locus normal obviously once operator origin osculating osculating plane P₁ parabola parallel perpendicular properties quaternion radius rectangular represents result right angles rotation S.aßy Saß scalar scalar equations second order self-conjugate shew solution sphere spherical straight line suppose surface surface of revolution tangent plane Taylor's Theorem tensor theorem three vectors triangle unit-vectors Vaß vector function versor w₁ write written Τρ φρ