question_answer

Let P, Q, R and S be the points on the plane with position vectors \[-\,2i-j,\]\[4i,\]\[3i+3j\] and \[-\,3i+2j\] respectively. The quadrilateral PQRS must be

A) parallelogram, which is neither a rhombus nor a rectangle

B) square

C) rectangle but not a square

D) rhombus but not a square

Correct Answer: A

Solution :

\[PQ=6i+j\], \[QR=-i+3j\], \[SR=6i+j\], \[PS=-i+3j\], \[PQ||BR\]and \[QR||PS\]

PQRS is a parallelogram

\[PQ.QR=\left( 6i+j \right)\left( -i+3j \right)\]

\[=-6+3\]

\[=-3\ne 0\]

\[\therefore \]      PQRS is not a rectangle.

\[|PQ|=\sqrt{36+1}=\sqrt{37}\]

\[|QR|=\sqrt{1+9}=\sqrt{10}\]

\[|PQ|\,\,\,\,=\,\,\,|QR|\]

\[\therefore \]It is not a rhombus.