Name _____________________
Worksheet 9.1
The Vector Cross Product
How do you find a vector which is perpendicular to two other vectors? Is there a product of two
vectors which is actually a vector? The answer to both these is yes – the vector cross product.
Please make note that this only works in three dimensions – unlike everything else we have
looked at the methods you are about to learn do not extend into higher dimensions easily, nor is
there a lower dimensional equivalent that you are already familiar with (such as the absolute
value being the 1-D norm).
1. Determinants
Before we dive into the cross product of two vectors, we need to learn about a little thing called
the determinant of a matrix (which will show up again before the year is done). Given a 3  3
matrix, the determinant is a single number, and is defined in a somewhat strange and mysterious
way.
a1
b1
c1
a2
b2
c2  a1
a3
b3
c3
Where
a
b
c
d
b2
c2
b3
c3
 b1
a2
c2
a3
c3
 c1
a2 b2
a3
b3
 ad  bc is the determinant of a 2  2 matrix.
Find the following determinants.
1
3
5
(a) 2 4 0 =
0
2
1
1 1 3
(b) 2
4
0=
3
3
3
2
6
(c) 1
0
1 3
10
4 =
5
a b c
(d) d
2a
e
f =
2b 2c
(e) What do you think it means if a determinant is zero?
2. Vector Cross Products
1 
0
0




ˆ
Now, recall the definition of unit vectors: iˆ  0 , ˆj  1 , k  0
 
 
 
0
0
1
 a   d  iˆ ˆj
The vector cross product of two vectors is defined as  b    e   a b
   
 c   f  d e
kˆ
c
f
1 
 2


Consider the vectors A  3 and B   4 
 
 
 2
 1 
(a) Find v  A  B (Remember that the answer will be a vector). Sketch all three vectors as best
as you can.
(b) Find u  B  A . Again, make a sketch.
(c) Is the cross product commutative? In other words, does A  B  B  A ? Check yourself!
(d) Find v  A and v  B
(e) What important fact have you discovered about the cross product of two vectors?
3. More Cross Products Properties
Show that each of these statements are true given the following vectors
1 
1
 3
 2
1 
0
0












ˆ
A   2 , B  1 , C  6 , D   1  , iˆ  0 , ˆj  1  , k  0
0 
1
0
 0 
0
0
1
(a) ( A  B)  A  0
(b) A  B  B  A (In other words, the magnitudes are equal)
(c) iˆ  ˆj  kˆ
(d) ˆj  iˆ  kˆ
(e) If A is parallel to C (verify by finding scalar multiple of A equal to C ), then A  C  0
(f) If A is perpendicular to D (use dot product to verify) then A  D  A D
4. Some Problems Involving Cross Products
 1
 2


(a) Find the cross product between the vectors 2 and  3 
 
 
 0 
1 
3
 2


(b) Find a vector perpendicular to both  1  and 1 
 1
 4
(c) Find the ordinary equation ( ax  by  cz  d ) of a plane containing the three points
H  (1, 2,  3), K  (2,  1, 4), and L  (1, 1, 1) Remember that you can use the cross product to
find a vector perpendicular to the plane containing all three of these points.