Calc 2 Lecture Notes
Section 10.4
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Section 10.4: The Cross Product
Big idea: Another meaningful way to define the multiplication of vectors is the “cross product.”
A cross product results in a vector for an answer, and is meaningful because it has a geometric
interpretation when the vectors being crossed are thought of as forming the sides of a
parallelogram.
Big skill: You should be able to calculate the cross product of two vectors, use the cross product
to find the area of a parallelogram, the shortest distance from a point to a line, the volume of a
parallelepiped, and the torque applied to an object.
Definition of the Cross Product in terms of Unit Vectors
The idea behind the cross product is that when two vectors are “multiplied” together, the answer
should be another vector. As far as the direction of the cross product vector, it was found to be
most useful if that direction were perpendicular to the two vectors being multiplied. So, if we
look at the cross product of the unit vectors i and j, we can see that the cross-product, written as
i  j , can have one of two answers: +k or –k. By convention, the answer of +k is chosen,
because it is the choice that obeys the right-hand rule.
Using these ideas that the answer must be perpendicular to the original vectors and the direction
must obey the right hand rule, we get the cross products in the table below. Also, if you try to
cross a vector with itself, there are an infinite number of perpendicular vectors, so the cross
product of a vector with itself is zero.
i j  k
j k  i
k i  j
j  i  k
k  j  i
i  k  j
ii  0
j j  0
k k  0
These cross products can be combined with the distributive property to compute the cross
product of an arbitrary vector:
 2i  3j  4k    i  2 j  2k 
  2i  3j  4k   i   2i  3j  4k    2 j   2i  3j  4k    2k 
  2i  i  3j  i  4k  i 
  2i   2 j  3 j   2 j  4k   2 j 
  2i   2k   3 j   2k   4k   2k  
 0  3  k   4 j   4k  0  8  i     4   j  6i  0 
 2i  8 j  7k
Calc 2 Lecture Notes
Section 10.4
Page 2 of 7
There is a more compact way to describe the calculations behind a cross product using
determinants. Recall that the determinant of a 22 matrix is:
Definition 4.1: Determinant of a 22 matrix
a1 a2
 a1b2  a2b1
b1 b2
And the determinant of a 33 matrix is:
Definition 4.2: Determinant of a 33 matrix
a1 a2 a3
b b
b b
b b
b1 b2 b3  a1 2 3  a2 1 3  a3 1 2
c2 c3
c1 c3
c1 c2
c1 c2 c3
This can be calculated faster using the trick:
Now that we have the review done, the cross product can be written as:
Definition 4.3: The 3-Dimensional Vector Cross Product
i
j k
a a3
a a
a a
a  b  a1 a2 a3  2
i  1 3 j 1 2 k
b2 b3
b1 b3
b1 b2
b1 b2 b3
Practice: Compute  2i  3j  4k    i  2 j  2k  using the matrix technique.
Calc 2 Lecture Notes
Section 10.4
Theorem 4.1: Zero Cross Products
For any vector a V3 , a  a  0 and a  0  0 .
Theorem 4.2: Cross Products are Othogonal
For any vectors a, b V3 , a  b is orthogonal to both a and b.
Theorem 4.3: Arithmetic Properties of The Cross Product
For vectors a, b, and c in V3, and any scalar d, the following hold:
(i).
(anticommutativity)
a  b  b  a
(ii).
d  a  b    da   b  a   db  (effect of scalar multiplication)
(iii).
(iv).
(v).
(vi).
a  b  c  a  b  a  c
a  b  c  a  c  b  c
a  b  c  a  b   c
a  b  c  a  c  b  a  b  c
(distributive property)
(distributive property)
(scalar triple product)
(vector triple product)
Theorem 4.4: Magnitude of the cross product
a  b  a b sin 
Proof:
2
2
2
2
a  b   a2b3  a3b2    a1b3  a3b1    a1b2  a2b1 
 a2 2b32  2a2 a3b2b3  a32b2 2
 a12b32  2a1a3b1b3  a32b12
 a12b2 2  2a1a2b1b2  a2 2b12
  a12  a2 2  a32  b12  b2 2  b32    a  b 
 a
2
 a
2
 a
2
b  a
2
b
b
2
2
2
b cos 2 
2
1  cos  
 sin  
2
2
 a  b  a b sin 
Corollary 4.1:
Two vectors are parallel if and only if a  b  0
2
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Calc 2 Lecture Notes
Section 10.4
Page 4 of 7
Application of Theorem 4.4: Area of a parallelogram:
Practice:
Find the area of the parallelogram with two adjacent sides formed by the vectors a = <3, 1, 4>
and b = <-2, 1, 8>.
Calc 2 Lecture Notes
Section 10.4
Application of Theorem 4.4: Distance from a point to a line:
Practice:
Find the distance from the line y = 2x – 5 to the point (-1, 4).
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Calc 2 Lecture Notes
Section 10.4
Page 6 of 7
Application of Theorem 4.4: Volume of a parallelipiped:
Note: c   a  b  
c1 c2
 a1 a2
b1 b2
c3
a3
b3
Practice:
Find the volume of the parallelepiped formed by the vectors a = 2i. b = 3j, c = j + k.
Calc 2 Lecture Notes
Section 10.4
Application of Theorem 4.4: Torque:
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