If the volume of the parallelopiped formed by the vectors i^+aj^+k^,j^+ak^ and ai^+k^ becomes minimum, then a=

AP EAMCET · Maths · Vector Algebra

If the volume of the parallelopiped formed by the vectors $\hat{i} + a \hat{j} + \hat{k}, \hat{j} + a \hat{k}$ and $a \hat{i} + \hat{k}$ becomes minimum, then $a =$

A $\frac{1}{3}$
B $\frac{1}{\sqrt{3}}$
C $\frac{2}{\sqrt{3}}$
D $\frac{2}{3}$

Verified Solution

Answer & Solution

Correct Answer

(B) $\frac{1}{\sqrt{3}}$

Step-by-step Solution

Detailed explanation

The Volume of parallelopiped is given by a→ b→ c→, where a→, b→, c→ represents the vectors that form the parallelopiped. Putting the given vectors, we get ⇒V=1a101aa01 =1-a-a2-a =1+a3-a For minimum value dVda=0…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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