If a = (1, 1, 0) and b = (1, 1, 1), then unit vector in the plane of a and b and perpendicular to a is

Let $$\mathbf{a}=\hat{\mathbf{i}}$$ and $$\mathbf{b}=\hat{\mathbf{j}}$$, the point of intersection of the lines $$\mathbf{r} \times \mathbf{a}=\mathbf{b} \times \mathbf{a}$$ and $$\mathbf{r} \times \mathbf{b}=\mathbf{a} \times \mathbf{b}$$ is

Which of the following vector is equally inclined with the coordinate axes?

If $$\hat{\mathbf{i}}+4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$$, and $$3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$$ are position vectors of $$A, B$$ and $$C$$ respectively and if $$D$$ and $$E$$ are mid points of sides $$B C$$ and $$A C$$, then $$\mathbf{D E}$$ is equal to