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

Question

Let P, Q, R and S be the points on the plane with position vectors -2 hat i - hat j , 4 hat i , 3 hat i +3 hat j and -3 hat i +2 hat j , respectively. The quadrilateral P Q R S must be a (A) parallelogram, which is neither a rhombus nor a rectangle (B) square (C) rectangle, but not a square (D) rhombus, but not a square

Tardigrade Admin · Accepted Answer

P Q=6 hati+ hatj Q R =- hati+3 hatj R S =-6 hati- hatj S P = hati-3 hatj |P Q| =√37=|R S| |Q R| =√10=|S P| P Q ⋅ Q R =-6+3=-3 ≠ 0 P Q is parallel to R S and their magnitudes are equal. ⇒ Quadrilateral P Q R S must be a parallelogram, which is neither a rhombus nor a rectangle.

Q. 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

square

rectangle, but not a square

rhombus, but not a square

PQ=6i^+j^​ QR=−i^+3j^​ RS=−6i^−j^​ SP=i^−3j^​ ∣PQ∣=37​=∣RS∣ ∣QR∣=10​=∣SP∣ PQ⋅QR=−6+3=−3=0 PQ is parallel to RS and their magnitudes are equal. ⇒ Quadrilateral PQRS must be a parallelogram, which is neither a rhombus nor a rectangle.

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