Proving volume of box using cross and dot product

[PAstudent]

Homework Statement

The diagram shows a box with parallel faces. Two of the faces are trapezoids and four of the faces are rectangles. The vectors A, B, and C lie along the edges as shown, and their magnitudes are the lengths of the edges. Define the necessary additional symbols and prove that the volume of the box is equal to AхB•C. Carefully drawn diagrams will enhance your presentation.

Homework Equations

The Attempt at a Solution

I know that the magnitude of the cross product is the area of a parallelogram. So the cross product of A and B would give the area of the rectangular faces. Then would the dot product give the volume because you already know the area of the rectangular faces?[/B]

 

[BvU]

You could work backwards a little: the magnitude of the cross product gives you ## |A| |B| ## so what do you need to get the volume ?
And where does ##\vec A \times \vec B ## point ?

 

[PAstudent]

So for the volume, I would need to find lCl. Wouldn't A x B point upwards and be perpendicular to A and B?

EDIT: the cross product gives the sin(theta) as well, would that have any importance?

 

[FeDeX_LaTeX]

The Attempt at a Solution

I know that the magnitude of the cross product is the area of a parallelogram. So the cross product of A and B would give the area of the rectangular faces. Then would the dot product give the volume because you already know the area of the rectangular faces?[/B]

As you've mentioned, the magnitude of the cross product of A with B gives the area of the base.

The height of the shape is the component of C normal to that base.

How can you combine these two pieces of information to find the volume?

 

[PAstudent]

Shouldn't the cross product give you another vector perpendicular? So wouldn't the cross product give the height of C since it would be a vector straight upward? So it would give you all the magnitudes of A,B,and C. Then are you saying I have to put that all into one equation?

 

[BvU]

No. (post #2): the cross product has nothing to do with C. It does indeed give a vector straight up. Makes some angle ##\gamma## with ##\vec C##.
And what about the dot product of ##\vec C## and ##\vec A \times \vec B## ?

 

[SammyS]

Homework Statement

The diagram shows a box with parallel faces. Two of the faces are trapezoids and four of the faces are rectangles. The vectors A, B, and C lie along the edges as shown, and their magnitudes are the lengths of the edges. Define the necessary additional symbols and prove that the volume of the box is equal to AхB•C. Carefully drawn diagrams will enhance your presentation.

View attachment 88405

Homework Equations

The Attempt at a Solution

I know that the magnitude of the cross product is the area of a parallelogram. So the cross product of A and B would give the area of the rectangular faces. Then would the dot product give the volume because you already know the area of the rectangular faces?[/B]

I don't see any trapezoids in this figure. If any pair of sides are trapezoids, then you need more information.

I do see a pair of parallelograms.

 

[vela]

I don't see any trapezoids in this figure. If any pair of sides are trapezoids, then you need more information.

I do see a pair of parallelograms.

Depending on your definition of trapezoid, a parallelogram could be considered a special case of a trapezoid.