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99 Percentile Qs Bank for JEE MainMathematicsVector Algebra

If a, b, c are distinct real numbers and P, Q, R are three points whose position vectors are respectively a i +b j +c k , b i +c j +a k and c i +a j +b k , then Q P R=

Options

  1. A⁻¹(a+b+c)
  2. B2
  3. C3
  4. D⁻¹ ( a^2+b^2+c^2 a b c )

Correct answer

C. 3

Step-by-step solution

Position vector of P=a i +b j +c k Position vector of Q=b i +c j +a k Position vector of R=c i +a j +b k Now, P Q = PV of Q- PV of P =(b i +c j +a k )-(a i +b j +c k ) =(b-a) i +(c-b) j +(a-c) k Similarly, P R =(c-a) i +(a-b) j +(b-c) k Now, P Q P R =[(b-a) i +(c-b) j +(a-c) k ] aligned & [(c-a) i +(a-b) j +(b-c) k ] = & (b-a)(c-a)+(c-b)(a-b)+(a-c)(b-c) = & a^2+b^2+c^2-a b-b c-c a | P Q |= & (b-a)^2+(c-b)^2+(a-c)^2 | P R |= & (c-a)^2+(a-b)^2+(b-c)^2 aligned Now, = P Q P R | P Q || P R | aligned & = a^2+b^2+c^2-a b-

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