Unformatted text preview: 87 1.6. CROSS PRODUCTS
and by deﬁnition
→→
− × − = 2A (△ABC )
v
w
→
=−
ı 11
2 −1 →
−− 21
1 −1 →
− 21
+k
.
12 The reasoning used in this example leads to the following general formula
for the cross product of two vectors in R3 from their components.
Theorem 1.6.5. The cross product of two vectors
→
→
− =x − +y − +z −
→
→
v
1ı
1
1k
→
→
− =x − +y − +z −
→
→
w
2ı
2
2k
is given by
→→→
− × − = − y1 z1
v
w
ı
y2 z2 →
−− x1 z1
x2 z2 →
− x1 y 1
+k
.
x2 y 2 Proof. Let P (x1 , y1 , z1 ) and Q(x2 , y2 , z2 ) be the points in R3 with position
→
→
vectors − and − , respectively. Then
v
w
→
−
→
→
→→
− × − = 2A (△OP Q) ı
v
w
= a1 − + a2 − + a3 k .
The three components of A (△OP Q) are its projections onto the three
coordinate directions, and hence by Proposition 1.6.4 each represents the
oriented area of the projection projP △OP Q of △OP Q onto the plane P
perpendicular to the corresponding vector.
Projection onto the plane perpendicular to a coordinate direction consists
→
−
of taking the other two coordinates. For example, the direction of k is
normal to the xy plane, and the projection onto the xy plane takes
P (x1 , y1 , z1 ) onto P (x1 , y1 ).
Thus, the determinant
x1 y 1
x2 y 2
represents twice the signed area of △OP3 Q3 , the projection of △OP Q
onto the xy plane, when viewed from above—that is, from the direction of
→
−
k —so the oriented area is given by
→
−
a3 k = 2A (△OP3 Q3 )
→
− x1 y 1
.
=k
x2 y 2 ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus, Vectors

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