Another method we can use to calculate the torque on a system is using the vector cross product.
The vector cross product is a mathematical operation that uses two vector inputs and provides a vectorial solution.
Let's say for example that we have two vectors that we want to cross, a and b. The resultant would be vector c.
We would represent the vector cross product with the following equation:
The magnitude of this cross-product would be:
The direction of the cross product would be determined from the right hand rule. To find the direction using the right hand rule, we start with our fingers in the direction of the first vector in the cross product (a in this case) and then curl our fingers in the direction of the second vector in the cross product (b in this case). The direction of our thumb tells us the direction of the resulting vector in the cross product.
IMPORTANT NOTE: The vector cross product is NOT commutative. The order in which the vector cross product is written is VERY important.
Practicing with the Vector Cross Product
Let's practice with the vector cross product a bit.
Let's look at different scenarios of the picture above and determine the cross product. Do this work on your own BEFORE checking the answer by clicking the box.
Scenario 1: What is a x b when the angle between the two is 0?
If we look at our equation for the magnitude of the vector from the cross product (a*b*sinθ), when theta is zero then sinθ is zero, thus making the cross product 0!
Before peaking at the solution to Scenario 1,
Check Here for a Hint!
How do we calculate the magnitude of the resulting vector with the cross product?
Scenario 2: What is a x b when the angle between the two is 90^{o}?
If we look at our equation for the magnitude of the vector from the cross product (a*b*sinθ), when theta is 90^{o} then sinθ is one, thus making the magnitude of the cross product a⋅b.
Scenario 3: What is a x b when the angle between the two is 180^{o}?
If we look at our equation for the magnitude of the vector from the cross product (a*b*sinθ), when theta is 180^{o} then sinθ is zero, thus making the cross product 0!
Scenario 4: What is a x b when the angle between the two is 270^{o}?
If we look at our equation for the magnitude of the vector from the cross product (a*b*sinθ), when theta is 270^{o} then sinθ is negative one, thus making the magnitude of the cross product -a⋅b.
Vector Cross Product of Unit Vectors
How could we determine the magnitude of the vector cross product of unit vectors?
Let's look at different scenarios of the picture above and determine the cross product. Do this work on your own BEFORE checking the answer by clicking the box.
Scenario 1: What is i x i?
The unit vector i is in the x-direction. But cross the unit vector i with itself, we are crossing two vectors that are in the same direction. Thus, the angle between the two vectors is zero. When the angle between the two vectors is zero, the magnitude of the cross product is zero!
Before peaking at the solution to Scenario 1,
Check Here for a Hint!
What is the angle between the two vectors we are crossing?
Scenario 2: What is j x j?
The unit vector j is in the y-direction. But cross the unit vector j with itself, we are crossing two vectors that are in the same direction. Thus, the angle between the two vectors is zero. When the angle between the two vectors is zero, the magnitude of the cross product is zero!
Scenario 3: What is k x k?
The unit vector k is in the z-direction. But cross the unit vector k with itself, we are crossing two vectors that are in the same direction. Thus, the angle between the two vectors is zero. When the angle between the two vectors is zero, the magnitude of the cross product is zero!
Scenario 4: What is i x j?
Using the right hand rule, if we start our fingers in the direction of i and curl them towards j, our thumbs points in the +k direction. So the cross product is +k.
Before peaking at the solution to Scenario 1,
Check Here for a Hint!
How does the right hand rule help us determine direction with the vector cross product?
Scenario 5: What is j x k?
Using the right hand rule, if we start our fingers in the direction of j and curl them towards k, our thumbs points in the +i direction. So the cross product is +i.
Scenario 6: What is k x i?
Using the right hand rule, if we start our fingers in the direction of k and curl them towards i, our thumbs points in the +j direction. So the cross product is +j.
Scenario 7: What is i x k?
Using the right hand rule, if we start our fingers in the direction of i and curl them towards k, our thumbs points in the -j direction. So the cross product is -j.
Scenario 8: What is k x j?
Using the right hand rule, if we start our fingers in the direction of k and curl them towards j, our thumbs points in the -i direction. So the cross product is -i.
Scenario 9: What is j x i?
Using the right hand rule, if we start our fingers in the direction of j and curl them towards i, our thumbs points in the -k direction. So the cross product is -k.
Here is a quick video demonstration as well:
Applying Vector Cross Product to Torque
We can calculate the torque on a system using the vector cross product as follows:
The magnitude of the vector cross product of r and F is:
The direction of the torque is determined using the vector cross product.
Other Methods for Calculating the Torque Using Cross Product
There are some other methods for calculating the cross product that we'd quickly like to highlight:
Using the cross product program on your graphing calculator: Your graphing calculator should have a program that allows you to calculate the vector cross product. You can find the instructions for using this program on a TI-83/84 calculator here. If you use a different type of calculator, you can use Google to search for instructions specific to your calculator.
Matlab: Matlab is a great resource for calculating the vector cross product. There is a specific function call 'cross' that allows you to calculate the vector cross product of two vectors. Here is a sample program for computing the cross product in Matlab:
Determinant Method: You can use the vector determinant method to find the cross product of two vectors (you will learn more about this method if you take linear algebra). With this method, we set up a 3x3 matrix. The top row of the matrix has the names of our unit vectors, the second row of the matrix contains our position vector, and the third row of our matrix contains our force vector (THEY MUST BE LISTED IN THIS ORDER). You then repeat that matrix a second time beside the first. Then, you draw diagonal lines as shown in the picture. Finally, you add the products of all the green lines and subtract the products of the red lines to get the resulting vector solution.
If you would like a little more information on how to use this method, the following video may be helpful.
Scalar method: A final way to compute the vector cross product is to compute the individual torque around each axis (x, y, and z) to represent the resultant vector. To use this method, you would use each of the following equations to get a final resulting torque: