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 \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.)<br /> (I suspect HallsofIvy&#039;s typo was due to a confusion over the symbols &quot; * &quot; and its synonym &quot; X &quot; for multiplication.)<br /> <br /> 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&#039;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 \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.)<br /> (I suspect HallsofIvy&#039;s typo was due to a confusion over the symbols &quot; * &quot; and its synonym &quot; X &quot; for multiplication.)<br /> <br /> 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&#039;s suggestion?

<br /> <br /> if X,Y, &amp; 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?