Page 181 - Advanced engineering mathematics

6.3 The Cross Product 161

To derive property (3), suppose both vectors are nonzero and recall that
cos(θ) = (F · G)/ F G , where θ is the angle between F and G. Now write
2
2
F G −(F · G) 2
2
2
2
2
2
= F G − F G cos (θ)
2
2
2
= F G sin (θ).
It is therefore enough to show that
2
2
2
2
F × G = F G −(F · G) ,
and this is a tedious but routine calculation.
Property (4) follows from (3), since two nonzero vectors are parallel exactly when the
angle θ between them is zero, and in this case, sin(θ) = 0. Properties (5) and (6) are routine
computations.
Property (4) provides a test for three points to be collinear, that is, to lie on a single line. Let
P, Q, and R be the points. These points will be collinear exactly when the vector F from P to Q
is parallel to the vector G from P to R. By property (4), this occurs when F × G = O.
One of the primary uses of the cross product is to produce a vector orthogonal to two given
vectors. This can be used to find the equation of a plane containing three given points. The
strategy is to pick one of the points and write the vectors from this point to the other two. The
cross product of these two vectors is normal to the plane containing the points. Now we know
a normal vector and a point (in fact three points) on the plane, so we can use equation (6.2) to
write the equation of the plane.
This strategy fails if the cross product is zero. But by property (4), this only occurs if the
given points are collinear, hence do not determine a unique plane (there are infinitely many planes
through any line in 3-space).

EXAMPLE 6.7
Find the equation of a plane containing the points P : (−1,4,2), Q : (6,−2,8), and
R : (5,−1,−1).
Use the three given points to form two vectors in the plane:
PQ = 7i − 6j + 6k and PR = 6i − 5j − 3k.
The cross product of these vectors is orthogonal to the plane of these vectors, so
N = PQ × PR = 48i + 57j + k
is a normal vector. By equation (6.2), the equation of the plane is

48(x + 1) + 57(y − 4) + (z − 2) = 0,
or
48x + 57y + z = 182.

SECTION 6.3 PROBLEMS

In each of Problems 1 through 4, compute F×G and G×F 2. F = 6i − k,G = j + 2k
and verify the anticommutativity of the cross product.
3. F = 2i − 3j + 4k,G =−3i + 2j
4. F = 8i + 6j,G = 14j
1. F =−3i + 6j + k,G =−i − 2j + k

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October 14, 2010 14:21 THM/NEIL Page-161 27410_06_ch06_p145-186