Calculating Triangle Area in 4-Space: Can Cross Products Be Used?

[CheesePlease]

Homework Statement

Find the area of triangle with vertices (-2,-2,2,2), (0,0,1,-1), (-1,-2,1,1)

Homework Equations

The Attempt at a Solution

The only way I know how to find the area of a triangle is by finding half the parallelogram. I.e. A = (1/2)||u x v||

But this requires cross product and you can't find the cross product in 4-space, can you?

Any help/explanation would be greatly appreciated.

 

[Mark44]

Homework Statement

Find the area of triangle with vertices (-2,-2,2,2), (0,0,1,-1), (-1,-2,1,1)

Homework Equations

The Attempt at a Solution

The only way I know how to find the area of a triangle is by finding half the parallelogram. I.e. A = (1/2)||u x v||

But this requires cross product and you can't find the cross product in 4-space, can you?

Any help/explanation would be greatly appreciated.

You'll need to form displacement vectors between pairs of vertices. You can then find the magnitudes of those displacement vectors. After that you can use the dot product to find the cosine of the angle between two adjacent sides of the triangle.

All three points lie in a plane (it happens to be in R⁴ but that's not important), so I would advise drawing a rough sketch of the three points. With a bit of trig you should be able to get the area of the triangle.

 

[CheesePlease]

Great, got it. Thanks you!