( a = i - j + k , b = i -2 j + k , c =p i +2 j +q k ) and ( d =p i +q j +2 k ) are given vectors. If the projection of ( c ) on ( a ) is (5 3 ) units and if a, ( b ) and ( c ) form a parallelopiped of volume 5 cubic units, then ( ⁻¹( b ). d ()= )

Step-by-step solution

Given, ( aligned & a = i - j + k , b = i -2 j + k & c =p i +2 j +q k , d =p i +q j +2 k aligned ) and projection of ( c ) on ( a =5 3 ) ( aligned & a c | a | =5 3 & p+q-2 3 =5 3 & p+q-2=15 & p+q=17 (i) aligned ) Also, given that ([ a b c ]=5 ) ( array lrl & [ array ccc 1 & -1 & 1 1 & -2 & 1 p & 2 & q array ]=5 & 1(-2 q-2)+1(q-p)+1(2+2 p)=5 & -2 q-2+q-p+2+2 p=5 & p-q=5 (ii) array ) By solving Eqs. (i) and (ii), we get (p=11, q=6 ) Now, ( aligned b d & =( i -2 j + k ) (p i +q j +2 k ) & =p-2 q+2 & =11-12+2 & =1 align