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 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.