Let w = i + j -2 k , and u and v be two vectors such that u v = w and v w = u . Let , , and t be real numbers such that u = i + j + k ,-t + + =0, -t + =0, and + -t =0. Match each entry in List-I to the correct entry in List-II and choose the correct option. LIST - I LIST - II (P) | v |^2 is equal to (1) 0 (Q) If = 3 ,then ^2 is equal to (2) 1 (R) If = 3 ,then ( + )^2 is equal to (3) 2 (S) If = 2 ,then t+3 is equal to (4) 3 (5) 5
A. ( P ) (2),( Q ) (1),( R ) (4),( S ) (5)
array lll u v = w & u v & u w \\ v w = u & v w & array aligned & w = i + j -2 k \\ & u = i + j + k aligned | array ccc - t & 1 & 1 \\ 1 & - t & 1 \\ 1 & 1 & - t array |=0 array ll u . w =0= + -2 & t=-1 or 2 \\ ( v . w ) v = u v & t=-1 \\ w | v |^2-0= w & + + =0 \\ | v |=1(P) & t=2 \\ & = = array aligned & Also, u v = w \\ & | u || v |=| w | \\ & | u |=| w |= 6 aligned Case-1: aligned & t =2 \\ & = = \\ & ^2+ ^2+ ^2=6 \\ & = 2 ,- 2 \\ & (S) t +3=5 aligned Case-2: aligned & t =-1 \\ & + + =0 \\ & + -2 =0 \\ & =0, =- \\ & ( Q ) = 3 , ^2=0, =- 3 \\ & ( R ) = 3 ( + )^2= ^2=3 \\ & ( P ) (2),( Q ) (1),( R ) (4),( S ) (5) aligned
Related: Mathematics — Vector Algebra · All PYQ Banks