Question

A unit vector coplanar with \(\hat{i}+\hat{j}+2\hat{k}\) and \(\hat{i}+2\hat{j}+\hat{k}\) and perpendicular to \(\hat{i}+\hat{j}+\hat{k}\) is

• A. \(\pm \frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})\)
• B. \(\pm \frac{1}{\sqrt{3}}(\hat{i}-\hat{j}+\hat{k})\)
• C. \(\pm \frac{1}{\sqrt{2}}(\hat{j}-\hat{k})\)
• D. \(\pm \frac{1}{\sqrt{2}}(\hat{j}+\hat{k})\)

Hint:

Write the vector as a combination of the two given vectors and force the dot product with (1,1,1) to be zero.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
A vector coplanar with \(\vec a\) and \(\vec b\) can be written as \(\alpha\vec a + \beta\vec b\).

Step 2: Set up
\(\vec a = \hat i + \hat j + 2\hat k\), \(\vec b = \hat i + 2\hat j + \hat k\). Their combination is \((\alpha + \beta)\hat i + (\alpha + 2\beta)\hat j + (2\alpha + \beta)\hat k\).
Perpendicular to \(\hat i + \hat j + \hat k\):
\[ (\alpha + \beta) + (\alpha + 2\beta) + (2\alpha + \beta) = 4\alpha + 4\beta = 0 \Rightarrow \beta = -\alpha \]

Step 3: Find the unit vector
The vector becomes \(\alpha(0\hat i - \hat j + \hat k)\), so the direction is \(\hat k - \hat j\), of magnitude \(\sqrt2\).
\[ \pm\frac{1}{\sqrt2}(\hat j - \hat k) \]
This is option (C). Option (A) is parallel to \(\hat i + \hat j + \hat k\), not perpendicular, and (D) is not in the plane.

Final Answer:
The unit vector is \(\pm\frac{1}{\sqrt2}(\hat j - \hat k)\), option (C). \[ \boxed{\pm\frac{1}{\sqrt{2}}(\hat{j} - \hat{k})} \]