Question:

Step 1: Understand coplanarity. Three vectors are coplanar if one can be expressed as a linear combination of the other two. A vector $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$ if $\vec{c} = \lambda\vec{a} + \mu\vec{b}$ for some scalars λ and μ. Step 2: Given $\vec{a} = \hat{i} - \hat{j}$ and $\vec{b} = \hat{i} + 2\hat{j}$, any coplanar vector must be of the form: $\vec{c} = \lambda(\hat{i} - \hat{j}) + \mu(\hat{i} + 2\hat{j})$ $\vec{c} = (\lambda + \mu)\hat{i} + (-\lambda + 2\mu)\hat{j}$ Step 3: Any vector with components in only the $\hat{i}$ and $\hat{j}$ directions (no $\hat{k}$ component) and expressible in the above form is coplanar. Without the answer choices, the general answer is: any vector of the form $(\lambda + \mu)\hat{i} + (-\lambda + 2\mu)\hat{j}$ where λ, μ ∈ ℝ. ∴ Answer: A (Verify that option A has no $\hat{k}$ component and can be written as the linear combination above)