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Unformatted text preview: 88 CHAPTER 1. COORDINATES AND VECTORS Similarly,
a1 − = 2A (△OP1 Q1 )
ı y1 z1
y2 z2 . Finally, noting that the direction from which the positive z -axis is
counterclockwise from the positive x-axis is −− , we have →
a2 − = 2A (△OP2 Q2 ) →
= −− x1 z1
x2 z2 . Adding these yields the desired formula.
In each projection, we used the 2 × 2 determinant obtained by omitting the
coordinate along whose axis we were projecting. The resulting formula can
be summarized in terms of the array of coordinates of − and −
x1 y1 z1
x2 y2 z2
by saying: the coeﬃcient of the standard basis vector in a given coordinate
direction is the 2 × 2 determinant obtained by eliminating the
corresponding column from the above array, and multiplying by −1 for the
We can make this even more “visual” by deﬁning 3 × 3 determinants.
A 3 × 3 matrix 17 is an array consisting of three rows of three entries
each, vertically aligned in three columns. It is sometimes convenient to
label the entries of an abstract 3 × 3 matrix using a single letter with a
double index: the entry in the ith row and j th column of a matrix A is
denoted 18 aij , giving the general form for a 3 × 3 matrix a11 a12 a13
A = a21 a22 a23 .
a31 a32 a33 We deﬁne the determinant of a 3 × 3 matrix as follows: for each entry
a1j in the ﬁrst row, its minor is the 2 × 2 matrix A1j obtained by deleting
17 Pronounced “3 by 3 matrix”
Note that the row index precedes the column index: aji is in the j th row and ith
column, a very diﬀerent place in the matrix.
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