MHT CET · Maths · Vector Algebra

The number of unit vectors perpendicular to \(\overline{\mathrm{a}}=(1,1,0)\) and \(\overline{\mathrm{b}}=(0,1,1)\) is

A one.
B two.
C three.
D infinite.

Verified Solution

Answer & Solution

Correct Answer

(B) two.

Step-by-step Solution

Detailed explanation

The vector perpendicular to \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\) is
\(\bar{a} \times \bar{b}=\left|\begin{array}{lll}
\hat{i} & \hat{j} & \hat{k} \\
1 & 1 & 0 \\
0 & 1 & 0
\end{array}\right|=\hat{i}-\hat{j}+\hat{k}\)
Since the length of this vector is \(\sqrt{3}\), the unit vector perpendicular to \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\) is \(\pm \frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{|\overline{\mathrm{a}} \times \overline{\mathrm{b}}|}= \pm \frac{1}{\sqrt{3}}(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})\)
Hence, there are two such vectors.

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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