The right-hand rule
For a "right-handed" coordinate system, the direction of the resultant vector for AxB can be determined as follows:
Point the forefinger of the right hand in the direction of A and point the second finger in the direction of B. The thumb will then point in the directionof the resultant vector.
The cross product is not commutative
If you think about this, you should realize that the cross product is not commutative. That is to say that AxB is not the same as BxA because thedirection of the resultant vector would not be the same.
Create a vector diagram on your graph board
Once again, in order for you to better understand the nature of a vector cross product, I recommend that you create a Cartesian coordinate system on yourgraph board, and draw the following two vectors.
A vector diagram for your graph board
Draw the first vector from the origin to a point at
x = 1y = 1.73
Label this vector A.Draw a second vector from the origin to a point at
x = 2.9y = 0.78
Label this vector B. The cross product
The cross product, AxB is defined as
AxB = Amag*Bmag*sin(angle)
where
- Amag is the magnitude of the vector A
- Bmag is the magnitude of the vector B
- angle is the angle between the two vectors, which must be less than or equal to180 degrees
The area of the parallelogram
Use the vectors that you have drawn on your graph board to construct a parallelogram and see if you can estimate the area of that parallelogram.
Even if you were a sighted student having the parallelogram drawn on high-quality graph paper, it would be something of a chore to manuallydetermine the area of the parallelogram.
Let's work through some numbers
Let's use the cross product to determine the area of the parallelogram.
Given the definition of the cross product, we see that there are three values that we need:
- Amag
- Bmag
- angle
Same vectors as before
If we were starting out with two new vectors, we could compute the magnitude of each vector using the Pythagorean theorem. We could also determine the angleby computing the vector dot product that I explained earlier in this module.
As you may have noticed, these are the same two vectors that we used earlier, and we computed those three values earlier. Going back and recovering thosethree values, we have
- Amag = 2.0
- Bmag = 3.0
- angle = 45 degrees (at least that is what I intended for it to be)
The area of the parallelogram
Using the earlier definition and the nomenclature for the Google calculator,
AxB = Amag*Bmag*sin(angle), or
AxB = 2.0*3.0*sin(45 degrees), or
AxB = 4.24 square units
The direction of the resultant vector
If you place the end of your thumb at the origin of your Cartesian coordinate system, you should be able, with reasonable comfort, to point your forefinger inthe direction of A and your second finger in the direction of B.
According to the right-hand rule , this means that the direction of the resultant vector is the direction that your thumb ispointing, or straight down into the graph board.
Perpendicular or parallel vectors
Now consider what happens as the angle varies between 90 degrees (perpendicular vectors) and 0 degrees (parallel vectors) for a given pair ofvectors.