Calculus: Analytic Geometry and Calculus, with Vectors

66 Vectors and geometry in three dimensions are determined by P and P as in the figure, we see that (1) IQPI =IQP'I =b+acos6 and ...

2.3 Scalar products, direction cosines, and tines in E, 67 2.3 Scalar products, direction cosines, and lines in E3 Let u and v b ...

68 Vectors and geometryin threedimensions long run, more sensible to concentrate upon the two formulas I(2.33) Jul IvI cos 0 u.v ...

2.3 Scalar products, direction cosines, and lines in E, 69 which provides an interesting way of making the more prosaic statemen ...

70 Vectors and geometryinthree dimensions The vector in parentheses is then the unit vector in the direction of P1P2, and its sc ...

2.3 Scalar products, direction cosines, and lines in E3 71 point-direction equations of lines because they reveal coordinates of ...

72 Vectors and geometry in three dimensions lies on the line PiP2 and hence that there is a number X for which P1P = AP1P2i show ...

2.3 Scalar products, direction cosines, and lines in E3 73 lying in the same plane, it is sometimes convenient to think of this ...

74 Vectors and geometry in three dimensions of checking the answer when (a) u =2i-3j+4k,v=2i+3j+4k (b) u = i + j, v = i (c)u=i+j ...

2.3 Scalar products, direction cosines, and lines in E3 75 Thus when we know about vectors, we can put the Newton law in the fol ...

76 Vectors and geometry in three dimensions when k = 1 and when k = 2. Remark: In case x2 = x'1, y2 Y1, 22 = zi, the transforms ...

2.3 Scalar products, direction cosines, and lines in E, 91 quite mysterious until it is realized that the lines P1P2 and P3P4 mi ...

78 Vectors and geometry in three dimensions direction and hence that there must be a positive constant A such that ok = Au,, for ...

2.4 Planes and lines in E, 79 A, B, C are not all 0, then the equation 14x+By+Cz+D=0 is the equation of a plane. In case .4 s 0, ...

80 Vectors and geometry in three dimensions where A (lambda) and μ (mu) are constants. Conversely, if A and μ are not both 0, th ...

2.4 Planes and lines in E, 81 Therefore, 473) dA(xi-x)+B(yi-y)+C(z,-z) (2. V/A2 + B2 + C2 This looks quite simple but, since P i ...

82 Vectors and geometry in three dimensions 6 Because several of the coordinates are zero, it is relatively easy to deter- mine ...

2.4 Planes and lines in E3 83 14 Determine the value of the parameter X for which the two planes which have the equations 2x + 3 ...

84 Vectors and geometry in three dimensions 21 Supposing that A, B, C are not all zero and that D2 0 D1, let ir1 and 1r2 be the ...

2.4 Planes and lines in E, 85 and a given point Po(xo,yo,zo). Hint: Consider separately the case in which Po is on L and the cas ...