Question involving cross product and planes

[mrcheeses]

Homework Statement

Take P;Q and R three points of R 3 not on the same line. If a = OP , b = OQ and c = OR are the position vectors corresponding to the three points, show that a x b + b x c + c x a is perpendicular to the plane containing P;Q and R

The Attempt at a Solution

I don't seem to understand how the cross product of these position vectors added up makes a vector perpendicular to the plane. To get a vector perpendicular to the plane, normally you would take these position vectors and subtract them to get a vector on the plane, do this with the other , and then take the cross product. If we just add vectors normal to each other, it just gives a scalar multiple of that vector, so in a way i think a x b is similar to a x b + b x c + c x a. If the cross product between just a and b are taken, a vector normal to the plane isn't the result.

 

[Dick]

Homework Statement

Take P;Q and R three points of R 3 not on the same line. If a = OP , b = OQ and c = OR are the position vectors corresponding to the three points, show that a x b + b x c + c x a is perpendicular to the plane containing P;Q and R

The Attempt at a Solution

I don't seem to understand how the cross product of these position vectors added up makes a vector perpendicular to the plane. To get a vector perpendicular to the plane, normally you would take these position vectors and subtract them to get a vector on the plane, do this with the other , and then take the cross product. If we just add vectors normal to each other, it just gives a scalar multiple of that vector, so in a way i think a x b is similar to a x b + b x c + c x a. If the cross product between just a and b are taken, a vector normal to the plane isn't the result.

No, a x b is not going to be a normal, as you said. You want to show the sum of the three terms is a normal. It will be a normal if it's perpendicular to (a-b) and (b-c), right? Use the dot product to check that.