Vector Refresher Part 4 Vector Cross Product Definition Vector Cross Product Definition Right Hand...
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Vector Refresher Part 4 Vector Cross Product Definition Vector Cross Product Definition Right Hand Rule Right Hand Rule Cross Product Calculation Cross
Vector Refresher Part 4 Vector Cross Product Definition Vector
Cross Product Definition Right Hand Rule Right Hand Rule Cross
Product Calculation Cross Product Calculation Properties of the
Cross Product Properties of the Cross Product
Slide 2
Cross Product The cross product is another method used to
multiply vectorsThe cross product is another method used to
multiply vectors
Slide 3
Cross Product The cross product is another method used to
multiply vectorsThe cross product is another method used to
multiply vectors Yields a vector resultYields a vector result
Slide 4
Cross Product The cross product is another method used to
multiply vectorsThe cross product is another method used to
multiply vectors Yields a vector resultYields a vector result This
vector is orthogonal to both vectors used in the calculationThis
vector is orthogonal to both vectors used in the calculation
Slide 5
Symbolism The cross product is symbolized with an x between 2
vectorsThe cross product is symbolized with an x between 2
vectors
Slide 6
Symbolism The following is stated Vector A crossed with vector
B.The following is stated Vector A crossed with vector B.
Slide 7
One Definition One definition of the cross product is
Slide 8
One Definition One definition of the cross product is x y
z
Slide 9
One Definition One definition of the cross product is x y z n
is a unit vector that describes a direction normal to both A and
B
Slide 10
One Definition One definition of the cross product is x y z n
is a unit vector that describes a direction normal to both A and B
Which way does it point?
Slide 11
Right Hand Rule The Right Hand Rule is used to determine the
direction of the normal unit vector. x y z
Slide 12
Right Hand Rule The Right Hand Rule is used to determine the
direction of the normal unit vector. x y z
Slide 13
Right Hand Rule The Right Hand Rule is used to determine the
direction of the normal unit vector. x y z Step 1: Point the
fingers on your right hand in the direction of the first vector in
the cross product
Slide 14
Right Hand Rule The Right Hand Rule is used to determine the
direction of the normal unit vector. x y z Step 1: Point the
fingers on your right hand in the direction of the first vector in
the cross product Step 2: Curl your fingers towards the second
vector in the cross product.
Slide 15
Right Hand Rule The Right Hand Rule is used to determine the
direction of the normal unit vector. x y z Step 1: Point the
fingers on your right hand in the direction of the first vector in
the cross product 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
Slide 16
One Definition This definition of the cross product is of
limited usefulness because you need to know the normal direction. x
y z
Slide 17
One Definition This definition of the cross product is of
limited usefulness because you need to know the normal direction. x
y z You can use this to find the angle between the 2 vectors, but
the dot product is an easier way to do this
Slide 18
Another DeFinition The cross product can also be evaluated as
the determinant of a 3x3 matrix
Slide 19
Another DeFinition The cross product can also be evaluated as
the determinant of a 3x3 matrix
Slide 20
Another DeFinition The cross product can also be evaluated as
the determinant of a 3x3 matrix
Slide 21
Another DeFinition The cross product can also be evaluated as
the determinant of a 3x3 matrix
Slide 22
Another DeFinition The cross product can also be evaluated as
the determinant of a 3x3 matrix
Slide 23
Evaluation of the Cross Product To evaluate this we start with
the term We start by crossing out the row and column associated
with i direction
Slide 24
Evaluation of the Cross Product To evaluate this we start with
the term This leaves a 2x2 matrix. The determinant of this matrix
yields the i term of the cross product
Slide 25
Evaluation of the Cross Product To evaluate this we start with
the term 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.
Slide 26
Evaluation of the Cross Product To evaluate this we start with
the term 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.
Slide 27
Evaluation of the Cross Product To evaluate this we start with
the term Next, we evaluate the j term. THERE IS AN INHERENT
NEGATIVE SIGN TO THIS TERM
Slide 28
Evaluation of the Cross Product To evaluate this we start with
the term 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
Slide 29
Evaluation of the Cross Product To evaluate this we start with
the term 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
Slide 30
Evaluation of the Cross Product To evaluate this we start with
the term Finally, we evaluate the k term.
Slide 31
Evaluation of the Cross Product To evaluate this we start with
the term Finally, we evaluate the k term. The determinant of the
remaining 2x2 matrix yields the k term of the cross product.
Slide 32
Evaluation of the Cross Product To evaluate this we start with
the term Finally, we evaluate the k term. The determinant of the
remaining 2x2 matrix yields the k term of the cross product.
Slide 33
Evaluation of the Cross Product To evaluate this we start with
the term The units of this vector will be the product of the units
of the vectors used to calculate the cross product.
Slide 34
Properties of the Cross Product Anti-commutative:
Slide 35
Anti-commutative: Not associative:
Slide 36
Properties of the Cross Product Anti-commutative: Not
associative: Distributive:
Slide 37
Properties of the Cross Product Anti-commutative: Not
associative: Distributive: Scalar Multiplication:
Slide 38
Other Facts about the Cross Product The cross product of 2
parallel vectors is 0
Slide 39
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
Slide 40
Example Problem Determine
Slide 41
First, we set up the matrix for the cross product evaluation
Determine
Slide 42
Example Problem To evaluate the i term, we need to disregard
the row and column i is found in. Determine
Slide 43
Example Problem Now, we take the determinant of the 2x2 matrix
that is left. Determine
Slide 44
Example Problem Multiply the diagonal that goes from the upper
left of the matrix to its lower right. Determine
Slide 45
Example Problem Subtract the product from the other diagonal to
complete the i term. Determine
Slide 46
Example Problem Remember that there is an inherent minus sign
in the j term. Determine
Slide 47
Example Problem The j term is found by disregarding the row and
column its found in, and taking the determinant of the remaining
2x2 matrix Determine
Slide 48
Example Problem The j term is found by disregarding the row and
column its found in, and taking the determinant of the remaining
2x2 matrix Determine
Slide 49
Example Problem The k term is found by disregarding the row and
column its found in, and taking the determinant of the remaining
2x2 matrix Determine
Slide 50
Example Problem The j term is found by disregarding the row and
column its found in, and taking the determinant of the remaining
2x2 matrix Determine
Slide 51
Example Problem The j term is found by disregarding the row and
column its found in, and taking the determinant of the remaining
2x2 matrix Determine
Slide 52
Example Problem Now we can simplify the equation Determine
Slide 53
Example Problem Now we can simplify the equation Determine
Slide 54
Example Problem Now we can simplify the equation Determine
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