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

Cross Product The cross product is another method used to multiply vectorsThe cross product is another method used to multiply vectors

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

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

Symbolism The cross product is symbolized with an x between 2 vectorsThe cross product is symbolized with an x between 2 vectors

Symbolism The following is stated “Vector A crossed with vector B.”The following is stated “Vector A crossed with vector B.”

One Definition One definition of the cross product is

One Definition One definition of the cross product is x y z θ

One Definition One 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 Definition One 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 Rule The Right Hand Rule is used to determine the direction of the normal unit vector. x y z θ

Right Hand Rule The Right Hand Rule is used to determine the direction of the normal unit vector. x y z θ

Right Hand Rule The 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 Rule The 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 Rule The 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 Definition This definition of the cross product is of limited usefulness because you need to know the normal direction. x y z θ

One Definition This 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 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

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

Another DeFinition The 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:

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 0 The magnitude of the cross product is equal to the area of a parallelogram bounded by 2 vectors

Example Problem Determine

First, we set up the matrix for the cross product evaluation Determine

Example Problem To evaluate the i term, we need to disregard the row and column i is found in. Determine

Example Problem Now, we take the determinant of the 2x2 matrix that is left. Determine

Example Problem Multiply the diagonal that goes from the upper left of the matrix to its lower right. Determine

Example Problem Subtract the product from the other diagonal to complete the i term. Determine

Example Problem Remember that there is an inherent minus sign in the j term. Determine

Example Problem The 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 Problem The 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 Problem The 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 Problem The 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 Problem The 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 Problem Now we can simplify the equation Determine

Example Problem Now we can simplify the equation Determine

Example Problem Now we can simplify the equation Determine