10.6: Vector Products of Two Vectors
We now have that long-awaited SECOND way to multiply vectors, so brace yourself. This multiplication process is called the "cross product," and it results in a vector answer, rather than a scalar one.
So basically, Vector C = Vector A X Vector B (the X represents "cross", so you'd read this as "vector C" is equal to Vector A cross Vector B). This process is NOT COMMUNITATIVE (authoritative use of capital letters, no?), so don't you dare assume that Vector B X Vector A will equal the same thing as Vector A X Vector B. If you do, by God, I will fight you. BUT, Vector A Cross Vector B WILL equal - Vector B Cross Vector A. Good to know.
Resultant Vector C will lie perpendicular to the plane that Vectors A and B reside on. What we must decipher is the direction in which Vector C is going; is it going toward us from the plane or away from us? We determine this by using the Right Hand Rule, and the steps for said rule are as follows
1) Cut a hole in the box
2) Put your ju.....oh wait, wrong steps, my bad.
1) For "Vector A Cross Vector B," Place your right-hand fingers along Vector A, and curl them towards Vector B
2) The direction in which your thumb is pointing represents the direction in which Vector C will travel.
3) For "Vector B Cross Vector A," simply do the opposite; place your fingers along Vector B, and curl them towards Vector A.
4) A good thing to remember is that if Vectors A and B are parallel, the product will be zero and the right hand rule consequently will fail miserably.
Other stuff to know:
Vector A Cross (Vector B + Vector C) = Vector A Cross Vector B + Vector A X Vector C
Here are the cross products of the unit coordinate vectors:
i x i=0 K x K=0 j x i=-k
k x i=j I x k=-j i x j=k
j x j=0 j x k=i k x j=-i
Example of Cross Products:
Find Vector C for Vector A Cross Vector B
Vector A= 8i+3j+4k
Vector A Cross Vector B= 72 i x i + 56 i x j + 16 i x k+ 27 j x i+ 21j x j + 6 j x k+ 36k x i + 28k x j + 8 k x k
Vector A Cross Vector B Reduced: 56k - 16j - 27k +6i +36j -28 i
Vector C = -22i +20j + 29k
Find Vector C for Vector A Cross Vector B Using Determinants:
Here is the basic rubric for determinants....so know it. It's important. I swear.
(i j k)
(Ax Ay Az)
(Bx By Bz)
Vector A Cross Vector B = (AyBz -AzBy)i - (AxBz - AzBx)j + (AxBy - AyBx)k
So then, let's apply it that rubric to determine Vector C (Using the same Vectors A and B)
(i j k)
(8 3 4)
(9 7 2)
Vector C = (6-28)i - (16 - 36)j + (56-27)k
Vector C = -22i + 20j +29k!
Alright, we're ALMOST done. Other stuff to know:
The Area of the parallelogram formed by Vectors A and B equals the absolute value of Vector A times the absolute value of Vector B times the Sine of the angle the two vectos form.
The Area of the triangle formed by Vectors A and B equals 1/2 times the absolute value of (Vector A times Vector B.)
And...THAT'S IT! Thank god.
My personalization.......IS LAAAAARRRYYYY!
Leo, you're next. Peace out.