The Cross Product of 2 Vectors
11.3
JMerrill, 2010
Unit Vectors in 2D
• In 2-D space, the unit vectors <0,1> and <1,0> are the standard unit
vectors and denoted by i = <1,0> and j = <0,1>
j = <0,1>
i = <1,0>
Unit Vectors in 2D
• Any vector can be written as a linear combination of the
vectors I and j.
• v = <v1, v2>
= v1<1,0> + v2<0,1>
= v1i + v2j
• The scalars v1 and v2 are the horizontal and vertical
components of v.
Writing a Linear Combination of Unit Vectors
• u has initial point (2, -5) and terminal point (-1,3). Write u
as a linear combination of the unit vectors i & j.
• u = <-1-2, 3-(-5)> = <-3, 8>
= -3i + 8j
Unit Vectors in 3D
The Cross Product
Finding The Cross Product
• An easy way to calculate the cross product is to use a
matrix. We use the determinant form with cofactor
expansion.
Finding The Cross Product
• An easy way to calculate the cross product is to use a
matrix. We use the determinant form with cofactor
expansion.
Finding the Cross Product
Subtraction sign
Addition Sign
Example
• Given u = i + 2j + k and
cross product of u x v.
i j k
u x v 1 2 1
3 1 2
v = 3i + j + 2k, find the
2 1
1 1
1 2

i 
j 
k
1 2
3 2
3 1
 (4  1)i  (2  3) j  (1  6)k
 3i  j  5k
You Try
• Given u = i + 2j + k and
cross product of v x u.
i
j
k
v x u  3 1 2
1 2 1
v = 3i + j + 2k, find the
1 2
3 2
3 1

i 
j 
k
2 1
1 1
1 2
 (1  4)i  (3  2) j  (6  1)k
 3i  j  5k
Using the Cross Product
• Find a unit vector that is orthogonal to both u = 3i – 4j + k
and
v = -3i + 6j.
• The cross product gives a vector that is orthogonal to both
u and v
=i -6i j– 3jk+ 6k
u x v  3 -4 1
-3 asks
6 for
0 a unit vector that’s orthogonal.
• The question
Using the Cross Product
• So, we need to divide by the magnitude of the orthogonal
vector.
• -6i – 3j + 6k
u x v  (6)2  ( 3)2  62  81  9
u x v
2
1
2

i  j  k
u x v
3
3
3
Triple Scalar Product
• Given 3 vectors u = 3i – 5j + k
v = 2j – 2k
w = 3i + j + k
• Find the volume of a parallelepiped having these vectors as
adjacent edges.
• The volume is found by
V = |u∙(v x w)|
Triple Scalar Product
3 -5
1
u  (v x w )  0 2 -2
3 1 1
2 -2
0 -2
0 2
3
 ( 5)
1
1 1
3 1
3 1
 3(4)  5(6)  1(6)
 36