Let a ,b ,c be three-unit vectors satisfying |a-b |^2+ |a-c |^2=10. Then Statement (I) : |a+2b |^2+ |2a+c |^2=2. Statement (II) : |2a+3b |^2+ |3a+2c |^2=10. Which of the above statements is (are) tru…

Question

Let a ,b ,c be three-unit vectors satisfying |a-b |^2+ |a-c |^2=10. Then Statement (I) : |a+2b |^2+ |2a+c |^2=2. Statement (II) : |2a+3b |^2+ |3a+2c |^2=10. Which of the above statements is (are) true?

Accepted Answer

Correct answer: D. (d) Given vector expression is |a-b |^2+ |a-c |^2=10 |a |^2+ |b |^2-2a b+ |a |^2+ |c |^2-2a c=10 ⇒-2a (b+c )+4=10 ⇒-2a (b+c )=6a (b+c)=-3 … … (i ) From statement (I): |a+2b |^2+ |2a+c |^2=2 Take L.H.S, ⇒ |a |^2+4 |b |^2+4a b+4 |a |^2+ |c |^2+4a c =1+4+4a (b+c )+4+1 =4 (-3 )+8+2 from (i ) =-12+10=-2 So, statement I is false. Now, take statement (II): |2a+3b |^2+ |3a+2c |^2 Take L.H.S. |2a+3b |^2+ |3a+2c |^2=4 |a |^2+9 |b |^2+12a b+9 |a |^2+4 |c |^2+12a c=4 1+9 1+12a b+9 1+4 1+12a c=12a (b+c )+26=12 -2+26=2 So statement II is also false.

$Let a, b, c be three-unit vectors satisfying | a - b |^{2} + | a - c |^{2} = 10. Then Statement (I) : | a + 2 b |^{2} + | 2 a + c |^{2} = 2 . Statement (II) : | 2 a + 3 b |^{2} + | 3 a + 2 c |^{2} = 10 . Which of the above statements is (are) true?$

Solution

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