Let the position vectors of points P ,   Q ,   R and S be a → = i ^ + 2 j ^ - 5 k ^ , b → = 3 i ^ + 6 j ^ + 3 k ^ , c → = 17 5 i ^ + 16 5 j ^ + 7 k ^ and d → = 2 i ^ + j ^ + k ^ , respectively. Then which of the following statements is true?
B. b → + 2 d → 3 is the position vector of a point which divides P R internally in the ratio 5 :   4
Given, P ( 1 , 2 , - 5 ) ,   Q ( 3 , 6 , 3 ) ,   R 17 5 , 16 5 , 7   &   S ( 2 , 1 , 1 ) Now solving options, Option A checking points are coplanar, P Q → = 2 , 4 , 8 ,   P R → = 12 5 , 6 5 , 12   &   P Q → = 1 , - 1 , 6 Now if points are coplanar then, P Q → P R → P S → = 0 Now solving L.H.S, we get, 2 4 8 12 5 6 5 12 1 - 1 6 = 2 5 1 2 4 12 6 60 1 - 1 6 = 2 5 96 - 24 - 72 = 0 Hence, they are coplanar So option A is wrong. Now solving, option B we get, b → + 2 d → 3 = 7 i + 8 j + 5 k 3 Now let point divides the P R   in   λ : 1 we get, So, on taking coefficient of i ^ we get, ⇒    17 λ 5 + 1 ( λ + 1 ) = 7 3 ⇒    17 λ 5 + 1 = 7 3 ( λ + 1 ) ⇒ 51 λ + 15 = 35 λ + 35 ⇒ 16 λ = 20 ⇒ λ = 5 4 , hence option B is correct and option C is wrong, Now solving option D we get, b → × d → = i ^ j ^ k ^ 3 6 3 2 1 1 = 3 i ^ + 3 j ^ - 9 k ^ ⇒ b → × d → 2 = 3 2 + 3 2 + 9 2 2 = 99 , Hence, option D is wrong.
Related: Mathematics — Vector Algebra · All PYQ Banks