Vector Refresher
Part 4
• Vector Cross Product
Definition
• Right Hand Rule
• Cross Product Calculation
• Properties of the Cross
Product
Cross Product
• The cross product is another method used to multiply
vectors
Cross Product
• The cross product is another method used to multiply
vectors
• Yields a vector result
Cross Product
• The cross product is another method used to multiply
vectors
• Yields a vector result
• This vector is orthogonal to both vectors used in the
calculation
Symbolism
• The cross product is symbolized with an x between 2
vectors
Symbolism
• The cross product is symbolized with an x between 2
vectors
• The following is stated “Vector A crossed with vector
B.” A ´ B
One Definition
One definition of the cross product is A ´ B = A B sin(q )nˆ
One Definition
One definition of the cross product is A ´ B = A B sin(q )nˆ
z
B
A
x
θ
y
One Definition
One definition of the cross product is A ´ B = A B sin(q )nˆ
z
n is a unit vector that
describes a direction
normal to both A and B
nˆ
B
A
x
θ
y
One Definition
One definition of the cross product is A ´ B = A B sin(q )nˆ
z
n is a unit vector that
describes a direction
normal to both A and B
Which way does it point?
B
A
y
θ
nˆ
x
Right Hand Rule
The Right Hand Rule is used to determine the direction
of the normal unit vector.
z
B
A
x
θ
y
Right Hand Rule
The Right Hand Rule is used to determine the direction
of the normal unit vector.
z
A´ B
B
A
x
θ
y
Right Hand Rule
The Right Hand Rule is used to determine the direction
of the normal unit vector.
z
A´ B
Step 1: Point the fingers on your
right hand in the direction of the first
vector in the cross product
B
A
x
θ
y
Right Hand Rule
The Right Hand Rule is used to determine the direction
of the normal unit vector.
z
A´ B
Step 1: Point the fingers on your
right hand in the direction of the first
vector in the cross product
B
A
x
θ
y
Step 2: Curl your fingers towards the
second vector in the cross product.
Right Hand Rule
The Right Hand Rule is used to determine the direction
of the normal unit vector.
z
A´ B
nˆ
Step 1: Point the fingers on your
right hand in the direction of the first
vector in the cross product
B
A
x
θ
y
Step 2: Curl your fingers towards the
second vector in the cross product.
Step 3: Your thumb points in the
normal direction that the cross
product describes
One Definition
This definition of the cross product A ´ B = A B sin(q )nˆ is
of limited usefulness because you need to know the
normal direction.
z
nˆ
B
A
x
θ
y
One Definition
This definition of the cross product A ´ B = A B sin(q )nˆ is
of limited usefulness because you need to know the
normal direction.
z
nˆ
B
A
x
θ
y
You can use this to find the
angle between the 2 vectors, but
the dot product is an easier way
to do this
Another DeFinition
The cross product can also be evaluated as the
determinant of a 3x3 matrix
Another DeFinition
The cross product can also be evaluated as the
determinant of a 3x3 matrix
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
U ´V =
Another DeFinition
The cross product can also be evaluated as the
determinant of a 3x3 matrix
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
iˆ
U ´V =
jˆ kˆ
Another DeFinition
The cross product can also be evaluated as the
determinant of a 3x3 matrix
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
iˆ
jˆ kˆ
U ´V = a b c
Another DeFinition
The cross product can also be evaluated as the
determinant of a 3x3 matrix
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
iˆ
jˆ kˆ
U ´V = a b
d e
c
f
Evaluation of the
Cross Product
To evaluate this we start with the iˆ term
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
We start by crossing out the row
and column associated with i
direction
iˆ
jˆ kˆ
U ´V = a b
d e
c
f
Evaluation of the
Cross Product
To evaluate this we start with the iˆ term
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
This leaves a 2x2 matrix. The
determinant of this matrix
yields the i term of the cross
product
iˆ
jˆ kˆ
U ´V = a b
d e
c
f
Evaluation of the
Cross Product
To evaluate this we start with the iˆ term
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
This leaves a 2x2 matrix. The
determinant of this matrix
yields the i term of the cross
product. Start by multiplying
the diagonal from the upper left
to the lower right.
iˆ
jˆ kˆ
U ´V = a b
d e
U ´V = (bf )iˆ
c
f
Evaluation of the
Cross Product
To evaluate this we start with the iˆ term
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
This leaves a 2x2 matrix. The
determinant of this matrix
yields the i term of the cross
product. Start by multiplying
the diagonal from the upper left
to the lower right. Now subtract
the product of the other
diagonal.
iˆ
jˆ kˆ
U ´V = a b
d e
U ´V = (bf - ce)iˆ
c
f
Evaluation of the
Cross Product
To evaluate this we start with the iˆ term
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
Next, we evaluate the j term.
THERE IS AN INHERENT
NEGATIVE SIGN TO THIS
TERM
iˆ
jˆ kˆ
U ´V = a b
d e
U ´V = (bf - ce)iˆ - () jˆ
c
f
Evaluation of the
Cross Product
To evaluate this we start with the iˆ term
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
Next, we evaluate the j term.
THERE IS AN INHERENT
NEGATIVE SIGN TO THIS
TERM. The determinant of the
remaining 2x2 matrix is
calculated in a similar fashion
iˆ
jˆ kˆ
U ´V = a b
d e
U ´V = (bf - ce)iˆ - (af ) jˆ
c
f
Evaluation of the
Cross Product
To evaluate this we start with the iˆ term
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
Next, we evaluate the j term.
THERE IS AN INHERENT
NEGATIVE SIGN TO THIS
TERM. The determinant of the
remaining 2x2 matrix is
calculated in a similar fashion
iˆ
jˆ kˆ
U ´V = a b
d e
c
f
U ´V = (bf - ce)iˆ - (af - cd) jˆ
Evaluation of the
Cross Product
To evaluate this we start with the iˆ term
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
Finally, we evaluate the k term.
iˆ
jˆ kˆ
U ´V = a b
d e
c
f
U ´V = (bf - ce)iˆ - (af - cd) jˆ
Evaluation of the
Cross Product
To evaluate this we start with the iˆ term
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
Finally, we evaluate the k term.
The determinant of the
remaining 2x2 matrix yields the
k term of the cross product.
iˆ
jˆ kˆ
U ´V = a b
d e
c
f
U ´V = (bf - ce)iˆ - (af - cd) jˆ + (ae)kˆ
Evaluation of the
Cross Product
To evaluate this we start with the iˆ term
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
Finally, we evaluate the k term.
The determinant of the
remaining 2x2 matrix yields the
k term of the cross product.
iˆ
jˆ kˆ
U ´V = a b
d e
c
f
U ´V = (bf - ce)iˆ - (af - cd) jˆ + (ae - bd)kˆ
Evaluation of the
Cross Product
To evaluate this we start with the iˆ term
U = aiˆ + bjˆ + ckˆ
V = diˆ + ejˆ + fkˆ
The units of this vector will be
the product of the units of the
vectors used to calculate the
cross product.
iˆ
jˆ kˆ
U ´V = a b
d e
c
f
U ´V = (bf - ce)iˆ - (af - cd) jˆ + (ae - bd)kˆ
Properties of the
Cross Product
Anti-commutative: A ´ B = -B ´ A
Properties of the
Cross Product
Anti-commutative: A ´ B = -B ´ A
Not associative:
(A ´ B)´ C ¹ A ´ (B ´C)
Properties of the
Cross Product
Anti-commutative: A ´ B = -B ´ A
Not associative:
(A ´ B)´ C ¹ A ´ (B ´C)
Distributive:
A ´ (B + C) = A ´ B + A ´ C
Properties of the
Cross Product
Anti-commutative: A ´ B = -B ´ A
Not associative:
(A ´ B)´ C ¹ A ´ (B ´C)
Distributive:
A ´ (B + C) = A ´ B + A ´ C
Scalar Multiplication:
k(A ´ B) = (kA)´ B = A ´ (kB)
Other Facts about
the Cross Product
The cross product of 2 parallel vectors is 0
Other Facts about
the Cross Product
The cross product of 2 parallel vectors is 0
The magnitude of the cross product is equal to the area
of a parallelogram bounded by 2 vectors
B
A
A
A = A´ B
Example Problem
Determine A ´ B
A = 3iˆ + 5 jˆ + kˆ
B = -iˆ - 2 jˆ + 3kˆ
Example Problem
Determine A ´ B
A = 3iˆ + 5 jˆ + kˆ
B = -iˆ - 2 jˆ + 3kˆ
iˆ
A´B =
First, we set up the matrix
for the cross product
evaluation
jˆ
kˆ
3 5 1
-1 -2 3
Example Problem
Determine A ´ B
A = 3iˆ + 5 jˆ + kˆ
B = -iˆ - 2 jˆ + 3kˆ
iˆ
A´B =
To evaluate the i term, we
need to disregard the row
and column i is found in.
jˆ
kˆ
3 5 1
-1 -2 3
Example Problem
Determine A ´ B
A = 3iˆ + 5 jˆ + kˆ
B = -iˆ - 2 jˆ + 3kˆ
iˆ
A´B =
Now, we take the
determinant of the 2x2
matrix that is left.
jˆ
kˆ
3 5 1
-1 -2 3
Example Problem
Determine A ´ B
A = 3iˆ + 5 jˆ + kˆ
B = -iˆ - 2 jˆ + 3kˆ
iˆ
A´B =
Multiply the diagonal that
goes from the upper left of
the matrix to its lower
right.
jˆ
kˆ
3 5 1
-1 -2 3
A ´ B = [(5)(3)]iˆ
Example Problem
Determine A ´ B
A = 3iˆ + 5 jˆ + kˆ
B = -iˆ - 2 jˆ + 3kˆ
iˆ
A´B =
Subtract the product from
the other diagonal to
complete the i term.
jˆ
kˆ
3 5 1
-1 -2 3
A ´ B = [(5)(3)- (1)(-2)]iˆ
Example Problem
Determine A ´ B
A = 3iˆ + 5 jˆ + kˆ
B = -iˆ - 2 jˆ + 3kˆ
iˆ
A´B =
Remember that there is an
inherent minus sign in the
j term.
jˆ
kˆ
3 5 1
-1 -2 3
A ´ B = [(5)(3)- (1)(-2)]iˆ -[] jˆ
Example Problem
Determine A ´ B
A = 3iˆ + 5 jˆ + kˆ
B = -iˆ - 2 jˆ + 3kˆ
iˆ
A´B =
jˆ
kˆ
3 5 1
-1 -2 3
A ´ B = [(5)(3)- (1)(-2)]iˆ -[(3)(3)] jˆ
The j term is found by
disregarding the row and
column it’s found in, and
taking the determinant of
the remaining 2x2 matrix
Example Problem
Determine A ´ B
A = 3iˆ + 5 jˆ + kˆ
B = -iˆ - 2 jˆ + 3kˆ
iˆ
A´B =
jˆ
kˆ
The j term is found by
disregarding the row and
column it’s found in, and
taking the determinant of
the remaining 2x2 matrix
3 5 1
-1 -2 3
A ´ B = [(5)(3)- (1)(-2)]iˆ -[(3)(3)- (1)(-1)] jˆ
Example Problem
Determine A ´ B
A = 3iˆ + 5 jˆ + kˆ
B = -iˆ - 2 jˆ + 3kˆ
iˆ
A´B =
jˆ
kˆ
The k term is found by
disregarding the row and
column it’s found in, and
taking the determinant of
the remaining 2x2 matrix
3 5 1
-1 -2 3
A ´ B = [(5)(3)- (1)(-2)]iˆ -[(3)(3)- (1)(-1)] jˆ
Example Problem
Determine A ´ B
A = 3iˆ + 5 jˆ + kˆ
B = -iˆ - 2 jˆ + 3kˆ
iˆ
A´B =
jˆ
kˆ
The j term is found by
disregarding the row and
column it’s found in, and
taking the determinant of
the remaining 2x2 matrix
3 5 1
-1 -2 3
A ´ B = [(5)(3)- (1)(-2)]iˆ -[(3)(3)- (1)(-1)] jˆ +[(3)(-2)]kˆ
Example Problem
Determine A ´ B
A = 3iˆ + 5 jˆ + kˆ
B = -iˆ - 2 jˆ + 3kˆ
iˆ
A´B =
jˆ
kˆ
The j term is found by
disregarding the row and
column it’s found in, and
taking the determinant of
the remaining 2x2 matrix
3 5 1
-1 -2 3
A ´ B = [(5)(3)- (1)(-2)]iˆ -[(3)(3)- (1)(-1)] jˆ +[(3)(-2)- (5)(-1)]kˆ
Example Problem
Determine A ´ B
A = 3iˆ + 5 jˆ + kˆ
B = -iˆ - 2 jˆ + 3kˆ
Now we can simplify the
equation
iˆ
A´B =
jˆ
kˆ
3 5 1
-1 -2 3
A ´ B = [(5)(3)- (1)(-2)]iˆ -[(3)(3)- (1)(-1)] jˆ +[(3)(-2)- (5)(-1)]kˆ
Example Problem
Determine A ´ B
A = 3iˆ + 5 jˆ + kˆ
B = -iˆ - 2 jˆ + 3kˆ
Now we can simplify the
equation
iˆ
A´B =
jˆ
kˆ
3 5 1
-1 -2 3
A ´ B = [(5)(3)- (1)(-2)]iˆ -[(3)(3)- (1)(-1)] jˆ +[(3)(-2)- (5)(-1)]kˆ
A ´ B = [15- (-2)]iˆ -[9 - (-1)] jˆ +[(-6)- (-5)]kˆ
Example Problem
Determine A ´ B
A = 3iˆ + 5 jˆ + kˆ
B = -iˆ - 2 jˆ + 3kˆ
Now we can simplify the
equation
iˆ
A´B =
jˆ
kˆ
3 5 1
-1 -2 3
A ´ B = [(5)(3)- (1)(-2)]iˆ -[(3)(3)- (1)(-1)] jˆ +[(3)(-2)- (5)(-1)]kˆ
A ´ B = [15- (-2)]iˆ -[9 - (-1)] jˆ +[(-6)- (-5)]kˆ
A ´ B =17iˆ -10 jˆ - kˆ