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The Normal Vector

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Scandal
#1
Jun8-09, 08:19 AM
P: 4
Find: AB X AC (vectors). Prove that the area of the triangle is [tex]\frac{3}{2}[/tex]

Further information
We're given the points A(1,0,0) , B(0,2,2), C(1,1,2)
So considering how this is the same question there is no doubt that we have to use the normal vector to find the answer.

The attempt at a solution
AB X AC = [2,2,-1]

|AB X AC| = [tex]\sqrt{2^{2}+2^{2}+(-1)^{2}}[/tex] = 3

Dividing this by 2 gives us the answer we're looking for. However, why are the length of the normal vector equal to the area of a cube? Are there any other special characteristics of the normal vector I should know before my oral exam in two days? The theme is geometry considered around third-dimension vectors and planes. All help is good help as I can't really catch the essence of the abstract normal vector.
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D H
#2
Jun8-09, 08:30 AM
Mentor
P: 15,055
You are looking for the area of a triangle, not a cube.

How does the cross product relate to the area of a parallelogram?
Scandal
#3
Jun8-09, 08:45 AM
P: 4
Might be I was unclear (perhaps of my poor English), but I found the answer, so this might not qualify for the homework help part of the forum, I don't know. The question is mostly why do the normal vector (cross product as many like to refer to it) have the properties of the area of a cube stretched by sides AB and AD (or in this case AC). In my perception it just a line which stretches outwards from the plane, 90 degrees in both 2D axises. I can't seem to comprehend why this line is so important, but somehow it is.

D H
#4
Jun8-09, 09:53 AM
Mentor
P: 15,055
The Normal Vector

I realize we have a language issue here. This is a cube:



A parallelogram is a plane figure:



The area of a parallelogram is the product of the lengths of any two adjacent sides and the sine of the angle between those sides:

[tex]A=ab\sin\theta[/tex]

This is also the magnitude of the cross product of the displacement vectors corresponding to those adjacent sides.



That the triangle has half the area of the parallelogram should be easy to visualize. A diagonal of a parallelogram splits the parallelogram into two identical triangles.
Scandal
#5
Jun8-09, 10:07 AM
P: 4
There it is! Thank you. I see now how cube was a wrong choice of words. What I meant to say was a square (3D vs. 2D), but as you say the correct would be a parallelogram. I think just the feeling of knowing somewhat what the normal vector is provides a lot when it comes to the presentation. I want to thank you again for the help.


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