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James Woods
Allen Dickerson
The Cross Product
In the start of the presentation on the cross product, we began by showing the
definition of the mathematical process. First we defined vector “a” to be a1i+a2j+a3k and
vector “b” to be b1i+b2j+b3k. From there we defined the cross between vectors “a” and
“b” to be:
a x b = (a2b3 – a3b2)i – (a1b3 –a3b1)j + (a1b2 –a2b1)k
Note: This is for 3-Dimensional space only and is not defined for 2-Dimensional
space
We next then showed how to calculate the determinant and arrive at the formula
shown in the definition. The best way to calculate the cross product of 2 vectors is by
using determinant form with cofactor expansion. We use cofactor expansion to find the
determinants by treating the vectors i, j, and k as if they were numbers filling in the 3x3
matrix. In return we do not actually get the determinant of the matrix; however we are
using the method of finding the determinant. We will now show this below.
This is how we find the determinant of the 2x2
matrix
After doing the first calculation all the way through using the determinant method in
the second line, we will end up with (a2b3 – a3b2)i – (a1b3 –a3b1)j + (a1b2 –a2b1)k which is
the same thing as shown above.
Next we showed that there are some specific algebraic properties pertaining to the
cross product. They are shown below.
1. a x b = -(b x a)
2. a x (b + d) = (a x b) + (a x d)
3.
4.
5.
6.
c(a x b) = (ca) x b = a x (cb) where c is a scalar in this case.
ax0=0xa=0
axa=0
a • (b x c) = b • (c x a) = c • (a x b)
After showing the algebraic properties we showed that there are also geometric
properties of the cross product.
1. a x b is orthogonal(perpindicular) to both vectors a and b if and only if the cross
product is not equal to 0
2. || a x b || = || a || || b || sin(theta)
3. a x b = 0 if and only if a and b are scalar multiples of each other
4. || a x b || = area of a parallelogram having a and b as adjacent sides
After showing the geometric properties, we showed why || a x b || is the area of the
parallelogram formed from the 2 adjacent vectors. We know that area is (base • height) so
in this case the base is || a || and the height is || b ||sin(theta). When these are multiplied
together you get || a || || b || sin(theta) which also equals || a x b || by the second geometric
property above. We also showed how to find the area of a triangle in space as well. We
know that the area of a triangle is ½ (base • height). From the area of a parallelogram we
know that the (base • height) equals || a x b ||. We then can conclude that the are of a
triangle in space is equal to ½ || a x b || (½ the area of a parallelogram) where a and b are
vectors in 3 dimensional space that are adjacent, creating the angle .
Below is the graphical representation of the cross product to show how the area is
calculated.
Lastly we showed what the triple scalar product was and what it can be used for.
The definition of the triple scalar product is
A • (B x C)= the determinant of the matrix show below
A • (B x C) = B • (C x A) = C • (A x B)
This formula comes in handy when trying to find the volume of a parallelepiped.
A parallelepiped is a parallelogram in three-dimensional space. The volume of such
figure is found by multiplying the area of the base by the height of the parallelepiped.
This looks like:
||projb x c a || • || b x c ||
In this case the projection is the height and the cross between b and c is the area of the
base and they are both multiplied together.
After some calculation and simplification, formula above can be reduced to the triple
scalar product A • (B x C)
http://tutorial.math.lamar.edu/Classes/CalcII/CrossProduct.aspx
http://mathworld.wolfram.com/ScalarTripleProduct.html
http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/cross
prod/crossprod.html