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If we carry out an identical procedure and eliminate z from the system of
equations, we have the following:
x
a 2 b 3 a 3 b 2
If we combine the two statements and set them equal to a constant k, we have
x
a 2 b 3 a 3 b 2
y
a 3 b 1 a 1 b 3
y
a 3 b 1 a 1 b 3
z
a 1 b 2 a 2 b 1
k
Note that we can make these fractions equal to k because every proportion can be
3
made equal to a constant k. (For example, if
then k could be
6 4 8 5 10 k,
1 10
1n
either or or any nonzero multiple of the form ) This expression gives us a
2 20
general form for a vector that is perpendicular to and The cross product,
a ! b ! , is defined to occur when k 1, and b ! a ! 2n .
a !
b ! .
occurs when k 1.
Formula for Calculating the Cross Product of Algebraic Vectors
k1a 2 b 3 is a vector perpendicular to
both a ! a 3 b
and b ! 2 , a 3 b 1 a 1 b 3 , a 1 b 2 a 2 b 1 2
, kR.
If k 1, then a !
If k 1, then b ! b !
a ! 1a 2 b 3 a 3 b 2 , a 3 b 1 a 1 b 3 , a 1 b 2 a 2 b 1 2
1a 3 b 2 a 2 b 3 , a 1 b 3 a 3 b 1 , a 2 b 1 a 1 b 2 2
It is not easy to remember this formula for calculating the cross product of two
vectors, so we develop a procedure, or a way of writing them, so that the memory
work is removed from the calculation.
Method of Calculating a ! b !
, where a ! 1a and b ! 1 , a 2 , a 3 2 1b 1 , b 2 , b 3 2
1. List the components of vector a ! in column form on the left side, starting with
a 2 and then writing a 3 , a 1 , and a below each other as shown.
2. Write the components of vector b ! 2
in a column to the right of a ! , starting with
b 2 and then writing and in exactly the same way as the components
of a ! b 3 , b 1 , b 2
.
3. The required formula is now a matter of following the arrows and doing the
calculation. To find the x component, for example, we take the down
product a 2 b 3 and subtract the up product a 3 b 2 from it to get a 2 b 3 a 3 b 2 .
(continued)
404 7.6 THE CROSS PRODUCT OF TWO VECTORS
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