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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 xaxis, yaxis, and z
axis. Note the following vectoralgebra identities: dot product ii=1,ij=o,i‘.12=o. Similarly
for 3 and 12 . cross product
ixi=5,ix3=h,3xi z~h;shnﬂady thrift. 1112
The 13,12  representation of a given vector A for a given Cartesian system. Represent A by an arrow with tailpoint 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= iA, a2 =3A, and a3 = iA. Thus we have the identity:
A =(Ai)i+(A3)}+(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. III3 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. [61]. 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 — (AB)C, and for the ﬁrst we have (C.A)B ~ (CB)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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