Question

using vector prove that if diagonals of a parallogram are equal it is a rectangle

Answer 1

Answer:Assuming vectors A and B are two adjacent sides of a parallelogram; a quick disproof of rhombus would be |A|≠|B|. However, assuming you want to check using the perpendicularity of the diagonals…

One diagonal will be A+B, the other will be A-B (or B-A, doesn’t matter for this method.) To check their perpendicularity, we can check to see if either a) their slopes are negative reciprocals (which would require knowledge of the vector coordinates) or b) the diagonals’ vectors are ±90° apart.

Since the dot product of two vectors — let’s use M and N, since A and B are already in use— is |M||N|cosθ, and cos(±90°)=0, we only need to show that the dot product of the diagonals is zero. Thus, if:

(A+B)⋅(A−B)=0

… then the diagonals are perpendicular, and the quadrilateral is a rhombus