Geometrical meaning of magnitude of vector product

[songoku]

TL;DR

I have notes that tells me one of the geometrical meaning of magnitude of vector product is related to perpendicular distance. Please see the diagram below

My notes says that the geometrical meaning of $$|\vec v \times \vec w | $$ is the perpendicular distance from point $V$ to line passing through $O$ and $W$ (all vectors are position vectors)

$$|\vec v \times \vec w | = |\vec v| |\vec w| \sin \theta$$

From the picture, the perpendicular distance is $|\vec v| \sin \theta$ but from the equation, there is extra $|\vec w|$ so it seems to me that the geometrical meaning is not the perpendicular distance but more like the multiplication of perpendicular distance from point $V$ to line $OW$ and magnitude of $\vec{OW}$

Am I misunderstood something?

Thanks

 

[mfb]

The magnitude of v x w is the area of the parallelogram. It's "related to" the height in the sense that the area is the height multiplied by |w|.

 

[songoku]

The magnitude of v x w is the area of the parallelogram. It's "related to" the height in the sense that the area is the height multiplied by |w|.

I understand the other geometrical meaning is area of parallelogram but is it correct to say that one of the geometrical meaning is (only) perpendicular distance from a point to a line (without stating there is multiplication with magnitude of the line) ?

Because if I am asked to find perpendicular distance from $V$ to line $OW$, I would calculate $|\vec v| \sin \theta$, not $|\vec v| |\vec w| \sin \theta$

Thanks

 

[FactChecker]

more like the multiplication of perpendicular distance from point $V$ to line $OW$ and magnitude of $\vec{OW}$

Am I misunderstood something?

You are correct. Your notes must have missed something.

 

[songoku]

Thank you very much mfb and FactChecker

 

[FactChecker]

So the magnitude of the cross product is the area of the parallelogram with the two vectors as adjacent sides.

 

[fresh_42]

Here (Introduction) is an interesting read about the various products:
https://arxiv.org/pdf/1205.5935.pdf

 

[robphy]

Here's another interesting article

Journal of Online Mathematics and Its Applications
Volume 6. June 2006. Article ID 1156
"The Geometry of the Dot and Cross Products"
Tevian Dray Corinne A. Manogue
https://www.maa.org/sites/default/files/images/images/upload_library/4/vol6/Dray2/Dray.pdf

 

[wrobel]

So the magnitude of the cross product is the area of the parallelogram with the two vectors as adjacent sides.

And what is interesting: the cross product is a (pseudo)vector and its magnitude should be of dimension L but not L^2 as it is so for the area :)

 

[robphy]

I understand the other geometrical meaning is area of parallelogram but is it correct to say that one of the geometrical meaning is (only) perpendicular distance from a point to a line (without stating there is multiplication with magnitude of the line) ?

Because if I am asked to find perpendicular distance from $V$ to line $OW$, I would calculate $|\vec v| \sin \theta$, not $|\vec v| |\vec w| \sin \theta$

Thanks

Use the unit vector \hat w along the line $OW$.
(Indeed, what role would the length of the segment $OW$ play
to find the distance from that line to point $V$?)