Calculus: Analytic Geometry and Calculus, with Vectors

86 Vectors and geometry in three dimensions Dividing by C and multiplying by -1 gives (6) 2(x-1)-(y-3)-(z-1)=0 or (7) 2x-y-z+2=0 ...

2.5 Determinants and applications 87 28 What can be said about the tip of the vector OP if OP = c,OPI + c2OP2 + c30P3, where cl, ...

8$ Vectors and geometry in three dimensions containing aik have been covered or removed. Thus for the determinant As, we have (2 ...

2.5 Determinants and applications 89 of the third row is demonstrated by the expansion 2 3 -1 6 1 3 7 3 -1^6 -2 = 3 3 0 0 0 3 7 ...

90 Vectors and geometry in three dimensions What this and similar formulas say is that we do not change the value of a determina ...

2.5 Determinants and applications 91 has a unique solution, the second system has no solutions, and the third system has many so ...

92 Vectors and geometry in three dimensions Now write formulas for the coefficients 11, B, C in the expansion A=f1(x-x1)+B(y-yi) ...

2.5 Determinants and applications 93 and hence that the formula for the volume of the tetrahedron can be put in the form Y= ±g y ...

94 Vectors and geometry in three dimensions This point P lies on ir if and only if D = 0, where (2) D = But (3) where x1 + X(xs ...

2.5 Determinants and applications 95 above and k is a scalar, we have (3) kP= -3k^2 k -k 3k k -k k/ P+Q=\ (^10) 0) When we multi ...

96 Vectors and geometry in three dimensions or the special case (10) (13 _ I 1 -') 1 2 2 0 1 -1C3) \-1 If we let (11) = (all a12 ...

2.6 Vector products and changes of coordinates in E3 97 so that z = Ay, y = Bx, and z = A(Bx). Show that z = Cx, where the matri ...

98 Vectors and geometry in three dimensions involving the angle 0 and the unit normal (thumb vector) n of Figure 2.612. We now w ...

2.6 Vector products and changes of coordinates in E, 99 neatly by writing i j kI uxv= 1 -2 31=5i+7j+3k. 12 -1 -11 We should all ...

100 Vectors and geometry in three dimensions hasty reading and preliminary ideas will be satisfactory preparation for future enc ...

2.6 Vector products and changes of coordinates in E3 101 coordinates of 0' are xo, yo, zo, so that (2.66) 00' = xoi + yoj + zok, ...

102 Vectors and geometry in three dimensions particular coordinate system is better than others and that there exist formulas re ...

2.6 Vector products and changes of coordinates in E3 103 So far as we know, some quadric surfaces may be rather complicated thin ...

104 Vectors and geometry in three dimensions 5 Show that the vectors U1=cosOsinOi+sinosinOj+cosOk u2= cosOcos Oi+sin Ocos Oj - s ...

2.6 Vector products and changes of coordinates in E; 105 11 Prove that if u and v and w are vectors having scalar components ul, ...