AP EAMCET2007MathematicsVector Algebra
Let a =a₁ i +a₂ j +a₃ k Assertion (A) : The identity | a i |^2+| a j |^2+| a k |^2=2| a |^2 holds for a . Reason (R) : a i =a₃ j -a₂ k , a j =a₁ k -a₃ i , a k =a₂ i -a₁ j Which of the following is correct?
Options
- ABoth ( A ) and ( R ) are true and ( R ) is the correct reason for (A)
- BBoth (A) and (R) are true but (R) is not the correct reason for (A)
- C(A) is true, (R) is false
- D(A) is false, (R) is true
Correct answer
A. Both ( A ) and ( R ) are true and ( R ) is the correct reason for (A)
Step-by-step solution
Given, a =a₁ i +a₂ j +a₃ k Now, | a i |^2=( a i ) ( a i ) aligned & = (a₃ j -a₂ k ) (a₃ j -a₂ k ) & =a₃^2+a₂^2 aligned aligned | a j |^2 & = (a₁ k -a₃ i ) (a₁ k -a₃ i ) & =a₁^2+a₃^2 aligned aligned | a k |^2 & = (a₁ i +a₁ j ) (a₂ i -a₁ j ) & =a₂ ^2+a₁ ^2 aligned Now, aligned | a i |^2 & +| a j |^2+| a k |^2 & =2 (a₁^2+a₂^2+a₃^2 ) & =2| a |^2 aligned Hence, both A and R are true and R is correct reason for A .