P Q R S is a parallelogram, where P Q =3 i +2 j -m k , P S = i +3 j + k and the area of the parallelogram is 90 . What is the value of m ?

Step-by-step solution

aligned & Area of parallelogram = 1 2 |( PQ PS )| & 90 = | array ccc i & j & k 3 & 2 & -m 1 & 3 & 1 array | & =[ i (2+3 m)- j (3+m)+ k (9-2)] & 90 = (2+3 m)^2+(3+m)^2+49 & = 4+12 m+9 m^2+9+m^2+6 m+49 & 90 = 10 m^2+18 m+62 aligned aligned & 90 = 10 m^2+18 m+62 & 10 m^2+18 m+62-90=0 & 10 m^2+18 m-28=0 & 5 m^2+9 m-14=0 & 5 m^2-5 m+14 m-14=0 & 5 m(m-1)+14(m-1)=0 & m- 14 5 , m=1 aligned