MHT CET · Maths · Vector Algebra

Let \(\overline{\mathrm{a}}, \overline{\mathrm{b}}\), and \(\overline{\mathrm{c}}\) be three non-zero vectors such that no two of these are collinear. If the vector \(\overline{\mathrm{a}}+2 \overline{\mathrm{~b}}\) is collinear with \(\overline{\mathrm{c}}\) and \(\overline{\mathrm{b}}+3 \overline{\mathrm{c}}\) is collinear with \(\overline{\mathrm{a}}\), then \(\overline{\mathrm{a}}+2 \overline{\mathrm{~b}}+6 \overline{\mathrm{c}}\) equals

A \(\lambda \bar{c}\) ( \(\lambda\) being some non-zero scalar)
B \(\lambda \bar{b}\) ( \(\lambda\) being some non-zero scalar)
C \(\lambda \bar{a}\) ( \(\lambda\) being some non-zero scalar)
D \(\overline{0}\) ( \(\lambda\) being some non-zero scalar)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\overline{0}\) ( \(\lambda\) being some non-zero scalar)

Step-by-step Solution

Detailed explanation

\(\overline{\mathrm{a}}+2 \overline{\mathrm{~b}}\) is collinear with \(\overline{\mathrm{c}}\)
\(\therefore \quad \bar{a}+2 \bar{b}=n \bar{c}\)...(i)
Similarly \(\overline{\mathrm{b}}+3 \overline{\mathrm{c}}=\mathrm{m} \overline{\mathrm{a}}\)...(ii)
\(m\) and \(n\) are non-zero scalars.
\(\therefore \quad\) (i) \(\Rightarrow \overline{\mathrm{a}}+2 \overline{\mathrm{~b}}+6 \overline{\mathrm{c}}=(\mathrm{n}+6) \overline{\mathrm{c}}\)
(ii) \(\Rightarrow \bar{a}+2 \bar{b}+6 \bar{c}=(2 m+1) \bar{a}\)
\(\Rightarrow \mathrm{n}+6=0\) and \(2 \mathrm{~m}+1=0\)
\(\Rightarrow \mathrm{n}=-6\) and \(\mathrm{m}=\frac{-1}{2}\)
\(\therefore \quad\) (i) \(\Rightarrow \bar{a}+2 \bar{b}+6 \bar{c}=0\)

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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