# Vectors and crossproducts

1. Sep 12, 2005

### SigmaCrisis

I'm given three points in 3D space (vectors), P, Q, and R. So I have to find a vector that is perpendicular to the plane formed by these points.

Anyone? Thanks.

I really just need some hint(s) for this.

2. Sep 12, 2005

### whozum

Draw two vectors from any of the points to other points. Only one plane defines those two vectors. The cross product of those two vectors is an orthogonal vector to that plane.

3. Sep 12, 2005

### SigmaCrisis

Umm, sorry, but could you elaborate? I don't quite get it...new to Calc III.

4. Sep 12, 2005

### mathwonk

if p,q are two points, subtracting Q-P gives a vector pointing from P to Q.

Thus if P,Q,R are 3 points, spanning a plane, the two vectors Q-P and Q-R are both parellel to that plane.

then there is a construction called cross product for finding sa vector perpendiculr to two given vectors. hence perpendiculr to the plane they are parallel to.

thus (Q-P) x (Q-R) is perpendicualr to the plane spanned by P,Q,R.

5. Sep 12, 2005

### SigmaCrisis

Thanks a bunch!

6. Sep 12, 2005

### SigmaCrisis

One more thing, I'm also asked to find the area of the triangle PQR.

7. Sep 12, 2005

### Staff: Mentor

Think about the magnitude of the cross product. What is the relationship to the area of a parallelogram with two sides (PQ and PR) given by the vectors?

Then what is the relationship of the area of the parallelogram to the triangle formed by the two sides, e.g. PQ, PR and the third QR?

8. Sep 13, 2005

### SigmaCrisis

Thanks a lot.