MHT CET202525 Apr 2025Evening ShiftMathematicsVector AlgebraActual
The altitude through vertex A of ABC with position vectors of points A , B , C as a , ~b , c respectively is
Options
- A| b c | | c - b |
- B| a b + b c + c a | | c - b |
- C| a b + b c + c a | | c b |
- D| b c | | a |
Correct answer
B. | a b + b c + c a | | c - b |
Step-by-step solution
Given vertices A, B, C with position vectors a , b , c , the area of ABC is expressed as Area = 1 2 |( b - a ) ( c - a )| . Expanding, ( b - a ) ( c - a ) = b c - b a - a c + a a . Since a a = 0 and using antisymmetry, this simplifies to a b + b c + c a . The area is also Area = 1 2 | c - b | h , where h is the altitude from A to BC. Equating the two expressions: 1 2 | c - b | h = 1 2 | a b + b c + c a | . Solving for h yields h = | a b + b c + c a | | c - b | , corresponding to option B .