Three points X, Y, Z have position vectors x,y,z. Show that X, Y and Z are collinear iff x ^ y + y ^ z + z ^ x = 0.
Here "^" denotes the vector cross product
If X, Y and Z are collinear then for some scalar . You can then verify that (remembering that the cross product of a vector with itself is always 0).
For the converse, if then (since the other two terms are 0). So is orthogonal to x. But it is also orthogonal to y and z. If x, y and z are linearly independent then is orthogonal to the whole space and is therefore 0. But that would mean that y and z are not linearly independent. That contradiction shows that the three vectors must be linearly dependent.
So one of them, z say, is a linear combination of the others, . Substitute that value for z into the equation and you will find that , which is the condition for X, Y and Z to be collinear.