Let ABC be a triangle of area 15 2 and the vectors AB = i ^ + 2 j ^ − 7 k ^ , BC = a i ^ + b j ^ + ck and AC = 6 i ^ + d j ^ − 2 k ^ , d 0 . Then the square of the length of the largest side

Question

Let ABC be a triangle of area 15 2 ​ and the vectors AB = i ^ + 2 j ^ ​ − 7 k ^ , BC = a i ^ + b j ^ ​ + ck and AC = 6 i ^ + d j ^ ​ − 2 k ^ , d > 0 . Then the square of the length of the largest side of the triangle ABC is _______

TestHub · Accepted Answer

​ Area = 2 1 ​ ​ i ^ 1 6 ​ j ^ ​ 2 d ​ k ^ − 7 − 2 ​ ​ = 15 2 ​ ( − 4 + 7 d ) i ^ − j ^ ​ ( − 2 + 42 ) + k ^ ( d − 12 ) ( 7 d − 4 ) 2 + ( 40 ) 2 + ( d − 12 ) 2 = 1800 50 d 2 − 80 d − 40 = 0 5 d 2 − 8 d − 4 = 0 5 d 2 − 10 d − 2 d − 4 5 d ( d − 2 ) + 2 ( d − 2 ) = 0 d = 2 or d = − 5 2 ​ ∵ d > 0 , d = 2 ( a + 1 ) i ^ + ( b + 2 ) j ^ ​ + ( c − 7 ) k ^ = 6 i ^ + 2 j ^ ​ − 2 k ^ a + 1 = 6& b + 2 = 2 , c − 7 = − 2 a = 5 b = 0 c = 5 ∣ A B ∣ = 1 + 4 + 49 ​ = 54 ​ ∣ BC ∣ = 25 + 25 ​ = 50 ​ ∣ A C ∣ = 86 + 4 + 4 ​ = 44 ​ ​ Ans. 54

Mathematics - Vector Question with Solution | TestHub

Doubts & Discussion