Vector question to think about

1. Sep 16, 2007

nick227

my teacher told me to think about this and i don't seem to get it. given vectors X,Y,& Z; is there geometric significance when X*(YxZ)=0

* is dot product and x is cross product

2. Sep 16, 2007

cornfall

geometry

Say X, Y and Z are orthonormal. Consider the part of X in the plane determined by the vectors Y and Z.

3. Sep 17, 2007

HallsofIvy

Staff Emeritus
The length of Y x Z is the area of a parallelogram with adjacent sides given by Y and Z.

X*(YxZ) is the the volume of a parallopiped having sides, at one vertex, given by X, Y, and Z. You can see that by using the formulas $X*Y= |X||Y|sin(\theta)$ and length of $X x Y= |X||Y|cos(\theta)$.

4. Sep 17, 2007

ice109

you shouldn't have just given him the answer

5. Sep 17, 2007

robphy

While HallsofIvy gave the interpretation when that special product is generally nonzero, there's still some interpretation left to do [for the OP] for the zero case.

Last edited: Sep 17, 2007
6. Sep 17, 2007

nick227

isn't $X x Y= |X||Y|sin(\theta)$ and $X*Y= |X||Y|cos(\theta)$?

also, from that definition, would the angle between Y and Z equal to 0? when i break the given down, i get:
|X||Y|cos(theta) x |X||Z|cos(theta).
i cant really see the geometric significance.

7. Sep 18, 2007

Math Jeans

Combinations of cross products and dot products like that are known as triple products.

8. Sep 18, 2007

nick227

wait so then if this triple product equals to 0, then does that mean the parallopiped is a cube?

9. Sep 18, 2007

robphy

You are (mostly) correct.
It is $| \vec X \times \vec Y | = |\vec X| |\vec Y| |\sin\theta|$ and $\vec X \cdot \vec Y = |\vec X| |\vec Y| \cos\theta$, where $\theta[/tex] is the angle between the vectors. ([itex] \vec X \times \vec Y$ is a vector with magnitude $|\vec X| |\vec Y| |\sin\theta|$ with direction perpendicular to the plane determined by $\vec X$ and $\vec Y$, according to the right-hand-rule.)
(I suspect HallsofIvy's typo was due to a confusion over the symbols " * " and its synonym " X " for multiplication.)

HallsofIvy gave the interpretation of the (scalar-)triple-product as the volume of a parallelopiped (a generally-slanted box with parallel sides) formed with those vectors. How would you describe this box if its volume were zero? What does that tell you about the relationship between $\vec X$, $\vec Y$ and $\vec Z$, along the lines of cornfall's suggestion?

10. Sep 18, 2007

nick227

if X,Y, & Z are orthonormal, than is X = (YxZ)? also, if the volume of the box is zero, then its not a 3d figure, its 2d. in that case, its a square.

11. Sep 18, 2007

robphy

So, what does that mean for vectors X, Y, and Z?

12. Sep 18, 2007

nick227

is it that all three vectors are on the same plane?

Last edited: Sep 18, 2007
13. Sep 18, 2007

robphy

Ok that's one way that the triple product is zero.
But suppose that X, Y, and Z are distinct nonzero vectors.
In fact, take a special case when X, Y, and Z are all vectors of length 1.
( When X,Y,Z are mutually orthogonal, you have a cube... with volume 1. )

Can you form a different parallelepiped with distinct nonzero vectors (with length 1) with a volume that is almost zero?... from there nudge things so that the volume is zero. What can you say about the vectors X,Y, and Z in that case? Now generalize to the general case.

14. Sep 18, 2007

robphy

ah... you changed your answer on me.

That's correct.

15. Sep 18, 2007

nick227

well i spent a lot of time thinking about it, and it finally clicked. Thanks for all the help!

16. Sep 19, 2007

HallsofIvy

Staff Emeritus
Yes to the last- that's exactly what I said. No to the first. X x Y is a vector, not a number and what you give is its length.

What you wrote makes no sense- you cannot take the the cross product of two numbers!
What does X*(YxZ)= 0 tell you about X and YxZ? What does that tell you, then, about X and both Y and Z?