Vector question to think about

[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

 

[cornfall]

geometry

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

 

[HallsofIvy]

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 $X x Y= |X||Y|cos(\theta)$.

 

[ice109]

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 $X x Y= |X||Y|cos(\theta)$.

you shouldn't have just given him the answer

 

[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.

 

[nick227]

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 $X x Y= |X||Y|cos(\theta)$.

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 can't really see the geometric significance.

 

[Math Jeans]

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

 

[nick227]

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

 

[robphy]

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

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 [itex]\theta[/tex] is the angle between the vectors. ($\vec X \times \vec Y$ 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?

 

[nick227]

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 [itex]\theta[/tex] is the angle between the vectors. ($\vec X \times \vec Y$ 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?

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.

 

[robphy]

if the volume of the box is zero, then its not a 3d figure, its 2d. in that case, its a square.

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

 

[nick227]

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

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

 

[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.

 

[robphy]

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

ah... you changed your answer on me.

That's correct.

 

[nick227]

ah... you changed your answer on me.

That's correct.

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

 

[HallsofIvy]

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 $X x Y= |X||Y|cos(\theta)$.

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

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.

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 can't really see the geometric significance.

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?