Question:

Step 1: Set up the general vector. Let the required unit vector be u . Since it must be coplanar with $\hat{i} - \hat{j}$ and $\hat{i} + 2\hat{j}$, it must be a linear combination of these vectors: $\mathbf{u} = \lambda(\hat{i} - \hat{j}) + \mu(\hat{i} + 2\hat{j}) = (\lambda + \mu)\hat{i} + (-\lambda + 2\mu)\hat{j}$ Step 2: Apply perpendicularity condition. For u to be perpendicular to the first vector $(\hat{i} - \hat{j})$: $\mathbf{u} \cdot (\hat{i} - \hat{j}) = 0$ $(\lambda + \mu)(1) + (-\lambda + 2\mu)(-1) = 0$ $\lambda + \mu + \lambda - 2\mu = 0$ $2\lambda - \mu = 0$ $\mu = 2\lambda$ Step 3: Substitute and simplify. Substituting $\mu = 2\lambda$ into the expression for u : $\mathbf{u} = (\lambda + 2\lambda)\hat{i} + (-\lambda + 4\lambda)\hat{j} = 3\lambda\hat{i} + 3\lambda\hat{j} = 3\lambda(\hat{i} + \hat{j})$ Step 4: Apply unit vector condition. For u to be a unit vector: $|\mathbf{u}| = 1$ $|3\lambda|\sqrt{1^2 + 1^2} = 1$ $|3\lambda|\sqrt{2} = 1$ $|\lambda| = \frac{1}{3\sqrt{2}}$ Step 5: Find the unit vectors. $\lambda = \pm\frac{1}{3\sqrt{2}}$ $\mathbf{u} = 3\left(\pm\frac{1}{3\sqrt{2}}\right)(\hat{i} + \hat{j}) = \pm\frac{1}{\sqrt{2}}(\hat{i} + \hat{j}) = \pm\frac{1}{2}\sqrt{2}(\hat{i} + \hat{j}) \cdot \frac{\sqrt{2}}{\sqrt{2}} = \pm\frac{1}{2}(\hat{i} + \hat{j})$ Verification: $|\pm\frac{1}{2}(\hat{i} + \hat{j})| = \frac{1}{2}\sqrt{2}\times\frac{\sqrt{2}}{\sqrt{2}} = \frac{1}{2}\sqrt{1+1} = \frac{\sqrt{2}}{2} \cdot \sqrt{2} = 1$ ✓ ∴ Answer: C