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.

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y

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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