Question

Find the volume of a pyramid with triangular base bounded by vectors (1,-1,2) and (1,1,1) and a vertex located at (3,2,5)?

Where shall I begin? I know the volume of the pyramid is 1/3 Base*height.

Answer 1

this is a linear algebra class, right?

Answer 2

yes

Answer 3

the magnitude of the cross product of two vectors can be interpreted as the area of a trapezoid

Answer 4

the base is triangular, so you want half of that
still thinking on how to get the height....

Answer 5

oh ok. right that makes sense. for the height would you just use a projection part?

P = vertex.

|P*(aXb)|/||aXb|| ?

Answer 6

that's what I was thinking :)
I just couldn't remember the projection formula.
Two heads are better than one!

Answer 7

yea haha ok. but the formula calls for the a 3rd vector from the origin to the vertex. P is just a point not a vector correct?

Answer 8

it can be interpreted as a the point at the tip of the vector... so it's kinda the same thing in this case, as you will see frequently in linear algebra and multi variable calc

Answer 9

it's the endpoint of the vector if the vector starts at the origin, to be precise

Answer 10

ok that makes sense.
thank you very much.

Answer 11

very welcome!