Vector Refresher Part 4 Vector Cross Product Definition Vector Cross Product Definition Right Hand...
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- Slide 1
- 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
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- Symbolism The following is stated Vector A crossed with vector
B.The following is stated Vector A crossed with vector B.
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- One Definition One definition of the cross product is
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- One Definition One definition of the cross product is x y
z
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- 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
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- 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?
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- 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
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- 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
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- 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
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- 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
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- Another DeFinition The cross product can also be evaluated as
the determinant of a 3x3 matrix
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- Another DeFinition The cross product can also be evaluated as
the determinant of a 3x3 matrix
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- 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
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- 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:
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- Anti-commutative: Not associative:
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- Properties of the Cross Product Anti-commutative: Not
associative: Distributive:
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- 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
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- First, we set up the matrix for the cross product evaluation
Determine
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- Example Problem To evaluate the i term, we need to disregard
the row and column i is found in. Determine
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- Example Problem Now, we take the determinant of the 2x2 matrix
that is left. Determine
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- Example Problem Multiply the diagonal that goes from the upper
left of the matrix to its lower right. Determine
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- Example Problem Subtract the product from the other diagonal to
complete the i term. Determine
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- Example Problem Remember that there is an inherent minus sign
in the j term. Determine
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- 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
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- Example Problem Now we can simplify the equation Determine
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- Example Problem Now we can simplify the equation Determine
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- Example Problem Now we can simplify the equation Determine