Given three points: [math]P(0,0,1)[/math], [math]Q(4,4,-1)[/math], [math]R(2,3,c)[/math], find the values of [math]c[/math] such that the angle at the vertex [math]R[/math] is a right angle. Answer: [math]c=\pm\sqrt{8}.[/math][br][br]Replace [math]Q(4,4,-1)[/math] with [math]Q(2,3,1)[/math] to see that there is no solution.[br][br]Set [math]Q=(2,3,-1)[/math] or, generally, [math]Q=(2,3,A)[/math], where [math]A\ne1[/math], to see that there is[br]one solution even though the dot product vanishes twice.