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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 - PowerPoint PPT Presentation
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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.”
One DefinitionOne definition of the cross product is
One DefinitionOne definition of the cross product is
x
y
z
θ
One DefinitionOne definition of the cross product is
x
y
z
θ
n is a unit vector that describes a direction normal to both A and B
One DefinitionOne definition of the cross product is
x
y
z
θ
n is a unit vector that describes a direction normal to both A and B Which way does it point?
Right Hand RuleThe Right Hand Rule is used to determine the direction of the normal unit vector.
x
y
z
θ
Right Hand RuleThe Right Hand Rule is used to determine the direction of the normal unit vector.
x
y
z
θ
Right Hand RuleThe Right Hand Rule is used to determine the direction of the normal unit vector.
x
y
z
θ
Step 1: Point the fingers on your right hand in the direction of the first vector in the cross product
Right Hand RuleThe Right Hand Rule is used to determine the direction of the normal unit vector.
x
y
z
θ
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.
Right Hand RuleThe Right Hand Rule is used to determine the direction of the normal unit vector.
x
y
z
θ
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
One DefinitionThis definition of the cross product is of limited usefulness because you need to know the normal direction.
x
y
z
θ
One DefinitionThis definition of the cross product is of limited usefulness because you need to know the normal direction.
x
y
z
θ
You can use this to find the angle between the 2 vectors, but the dot product is an easier way to do this
Another DeFinitionThe cross product can also be evaluated as the determinant of a 3x3 matrix
Another DeFinitionThe cross product can also be evaluated as the determinant of a 3x3 matrix
Another DeFinitionThe cross product can also be evaluated as the determinant of a 3x3 matrix
Another DeFinitionThe cross product can also be evaluated as the determinant of a 3x3 matrix
Another DeFinitionThe cross product can also be evaluated as the determinant of a 3x3 matrix
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
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
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.
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.
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
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
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
Evaluation of the Cross Product
To evaluate this we start with the term
Finally, we evaluate the k term.
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.
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.
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.
Properties of the Cross Product
Anti-commutative:
Properties of the Cross Product
Anti-commutative:Not associative:
Properties of the Cross Product
Anti-commutative:Not associative:Distributive:
Properties of the Cross Product
Anti-commutative:Not associative:Distributive:Scalar Multiplication:
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 0The magnitude of the cross product is equal to the area of a parallelogram bounded by 2 vectors
Example ProblemDetermine
Example ProblemFirst, we set up the matrix for the cross product evaluation
Determine
Example ProblemTo evaluate the i term, we need to disregard the row and column i is found in.
Determine
Example ProblemNow, we take the determinant of the 2x2 matrix that is left.
Determine
Example ProblemMultiply the diagonal that goes from the upper left of the matrix to its lower right.
Determine
Example ProblemSubtract the product from the other diagonal to complete the i term.
Determine
Example ProblemRemember that there is an inherent minus sign in the j term.
Determine
Example ProblemThe j term is found by disregarding the row and column it’s found in, and taking the determinant of the remaining 2x2 matrix
Determine
Example ProblemThe j term is found by disregarding the row and column it’s found in, and taking the determinant of the remaining 2x2 matrix
Determine
Example ProblemThe k term is found by disregarding the row and column it’s found in, and taking the determinant of the remaining 2x2 matrix
Determine
Example ProblemThe j term is found by disregarding the row and column it’s found in, and taking the determinant of the remaining 2x2 matrix
Determine
Example ProblemThe j term is found by disregarding the row and column it’s found in, and taking the determinant of the remaining 2x2 matrix
Determine
Example ProblemNow we can simplify the equation
Determine
Example ProblemNow we can simplify the equation
Determine
Example ProblemNow we can simplify the equation
Determine
