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s_notes03_v2 - III—l CARTESIAN COORDINATE SYSTEM Three...

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Unformatted text preview: III—l CARTESIAN COORDINATE SYSTEM: Three mutually perpendicular axes are chosen: -through a common point (“origin”) -each axis a number line with zero at origin -right—handed labels (usually X, y, 2) -location of origin arbitrary ~directions of axes arbitrary, (subject to mutual perpendicularity) Coordinates for any given point P in E3 ~unique triple of real numbers obtained by taking unique projection of P on each axis -values depend on choice of origin and of axis directions The unit vectors i , 3, 12 for a given coordinate system The three unit vectors with positive directions of, respectively, the x-axis, y-axis, and z- axis. Note the following vector-algebra identities: -dot product i-i=1,i-j=o,i‘.12=o. Similarly for 3 and 12 . -cross product ixi=5,ix3=h,3xi z~h;shnflady thrift. 111-2 The 13,12 - representation of a given vector A for a given Cartesian system. Represent A by an arrow with tail-point at the origin. Let (a1, a2, a3) be the coordinates of the head point P. Take an arrow for ali with tail at the origin, then an arrow for azj with tail at the head of the arrow for ali, then an arrow for 33k with tail at the head of the arrow for afl. By geometry, the head point of the last arrow is P. Conclusion: A = a1i+ a2} + an} by the definition of vector addition. .. Note that a1= i-A, a2 =3-A, and a3 = i-A. Thus we have the identity: A =(A-i)i+(A-3)}+(A.12)£. This is an example of the frame identity . The coordinate formulas for our four basic vector operations now follow, by using the laws of vector algebra, as shown in the text. A frame in E3 is a set of three mutually perpendicular unit vectors. Given a chosen origin point, a frame can be used to define axes for a new Cartesian coordinate system at that origin. Triple products are special combinations of the dot and cross products which have useful properties for simplifying expressions of vector algebra. See §3.6. III-3 TRIPLE PRODUCTS Scalar triple product: Has form A-(Bx C)01‘(AxB)-C. Using coordinate formulas for operations, we can show that these two expressions always have the same value and that that value is a scalar. Let [ABC] abbreviate the second expression. We can also conclude: [ABC] = [BCA] = [CAB] = -——[BAC] = - [ACB] == ~[CBA]. Furthermore, the absolute value I [ABC]| is equal to the volume of the parallelpiped determined by taking arrows for A, B. C with a common tail point. See text fig. [6-1]. Vector triple product: Has form (A x B) x c or A x (B x C) . These two expressions rarely have the same value. For the second expression, we have the value (A'C)B -— (A-B)C, and for the first we have (C.A)B ~ (C-B)A. Note that the first value is a vector lying in the plane of A and B, while the second lies in the plane of B and C The triple products are especially useful for simplifying expressions of vector algebra. See for example, the simplifications in the text for (A x B) -(c x D) and for (A x B) x (c x D) . (The “scalar quadruple product” and the “vector quadruple product”.) ...
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