COMEDK · Maths · 33. Vector Algebra

\(\text { If } \vec{a} \text { and } \vec{b} \text { are unit vectors, then the angle between } \vec{a} \text { and } \vec{b} \text { for which } a-\sqrt{2} \vec{b} \text { is a unit vector is }\)

A \(\dfrac{\pi}{3}\)
B \(\dfrac{\pi}{6}\)
C \(\dfrac{\pi}{4}\)
D \(\dfrac{\pi}{2}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\dfrac{\pi}{4}\)

Step-by-step Solution

Detailed explanation

Given that \(\vec{a}\) and \(\vec{b}\) are unit vectors, we have \(|\vec{a}| = 1\) and \(|\vec{b}| = 1\). The vector \(\vec{a} - \sqrt{2}\vec{b}\) is a unit vector, which implies \(|\vec{a} - \sqrt{2}\vec{b}| = 1\). Squaring both sides, we get…

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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From COMEDK

Find the correct matches of the substances, given in Column I, from their characteristic properties given in Column II.

S.No.	Column I	S.No.	Column II
W	\( \mathrm{Tl} \)	A	Decomposes steam to form its dioxide and dihydrogen gas.
X	\( \mathrm{CCl}_4 \)	B	Consists of twenty 6 -membered rings and twelve 5 membered rings.
Y	\(\mathrm{Tin}\)	C	+3 oxidation state is highly oxidising.
Z	\( \mathrm{C_{60}} \)	D	Electron precise species, neither acts as electron donor nor as electron acceptor.

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