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 B=9i+7j+2k





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.

Website: http://omega.albany.edu:8008/calc3/cross-product-dir/cornell-lecture.html





My personalization.......IS LAAAAARRRYYYY!






Leo, you're next. Peace out.