This is a homework questions I've spent a lot of time trying to solve and I am not sure if I am doing it right. Here is the question:

 

 

A triangular prism has a base defined by the points (1,3,0), (3,-4,0) and (-2,1,0). The prism has a slant height given by the vector (2,3,7). Determine the volume of this prism.

 

 

So far I've gotten three answers doing this questions three different ways (using dot product, cross product, cosine law, etc.):

 

98.43 units³

160 units³

92.14 units³

 

Can someone please explain to me how I can solve this problem? This is for a gr. 12 calculus and vectors course. Thank you!

Edited March 10, 201114 yr by Twinbird24

The following is not advice on how to solve this problem directly.

 

Maybe it would be easier for you to understand if you first chose a vertice and translated it to the origin followed by rotating the vertices so that the prism would sit aligned to the axis. This would naturally require some extra work on your part but might allow you to more clearly see the answer for yourself.

 

It may also help you to visualize this if you do it in a 3D modeling studio, but again these are just ideas.

Edited March 10, 201114 yr by Xittenn

Prism Volume = [math]\frac{1}{2} \times length \times width \times height[/math]

To be a bit more explicit.

1. The volume of any standard right prism is area of base x height.

Thus the volume of a triangular prism is area of triangle x height

2. I am presuming that you know how to calculate the area of the triangle that the base is formed from

Calculate this area in units²

3. The volume of a slant prism needs some thinking about

Imagine a right prism that has edges and faces that at are right-angles to the plane upon which the base is formed

Make small slices parallel to the plane (imagine a packet of biscuits for a cyclindrical prism)

Push each slice a little to the side compared to the one below - and what have you got?

Now you should be able to guess what your third element must be and how to calculate it from the vector

4. Wack em all together