11.3 The Cross Product
Denition. The cross product of two nonzero vectors a and b is the vector
a × b = (|a||b| sin θ) n
where θ is the angle between the vectors a and b, 0 ≤ θ ≤ π and n is a unit vector
perpendicular to both a and b and whose direction is given by the right-hand rule:
If the ngers of your right hand curl through the angle θ from a to b, then your thumb
points in the direction of n.
If a = 0 or b = 0, then a × b = 0.
Properties of the cross product
1. a × b is orthogonal to both a and b.
2. a × b = −b × a
3. |a × b| = |a||b| sin θ = the area of the parallelogram determined by a and b
4. Two nonzero vectors a and b are parallel if and only if a × b = 0.
Cross product in component form
A determinant of order 2 is dened by
a b
= ad − bc
c d
A determinant of order 3 is dened by
a1 a2 a3
b b
b b
b b
b1 b2 b3 =a1 2 3 − a2 1 3 + a3 1 2
c1 c2
c1 c3
c2 c3
c1 c2 c3
=a1 (b2 c3 − b3 c2 ) − a2 (b1 c3 − b3 c1 ) + a3 (b1 c2 − b2 c1 )
The cross product of a = a1 i + a2 j + a3 k and b = b1 i + b2 j + b3 k is
i j k
a a
a a
a a
a × b = a1 a2 a3 =i 2 3 − j 1 3 + k 1 2
b2 b3
b1 b3
b1 b2
b1 b2 b3
=(a2 b3 − a3 b2 )i − (a1 b3 − a3 b1 )j + (a1 b2 − a2 b1 )k
or
a × b = ha2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 i
Example 1. Find the area of the triangle determined by the points A(1, 0, 0), B(1, 1, 1)
and C(2, −1, 3).
Example 2. Find a unit vector perpendicular to the plane that contains the points
A(1, 0, 0), B(1, 1, 1) and C(2, −1, 3).
Scalar triple product
For three vectors a, b and c, we dene their scalar triple product as
a · (b × c)
Facts
1. If a = ha1 , a2 , a3 i, b = hb1 , b2 , b3 i and c = hc1 , c2 , c3 i, then
a1 a2 a3
a · (b × c) = b1 b2 b3
c1 c2 c3
2. |a·(b×c)| = the volume of the parallelepiped determined by the vectors a, b and c.
3. Three nonzero vectors a, b and c are coplanar (i.e., they lie in the same plane) if
and only if a · (b × c) = 0.
Example 3. Determine if the vectors a = h0, 1, 1i, b = h1, 4, −7i and c = h2, −1, 4i
are coplanar. If they are not coplanar then nd the volume of the parallelepiped that
has the given vectors as adjacent edges.