Vectors — common mistakes

When finding the intersection of two lines, $\mathbf{r}_1 = \mathbf{a} + \lambda\mathbf{d}_1$ and $\mathbf{r}_2 = \mathbf{b} + \mu\mathbf{d}_2$, you will set up three simultaneous equations (for $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$). Solve for $\lambda$ and $\mu$ using two of the equations. You MUST substitute these values back into the third equation. If it holds true, the lines intersect. If not, they are skew (assuming they are not parallel).

A position vector gives the coordinates of a point relative to the origin, like an address. A direction vector isn't fixed to the origin; it just describes a direction and magnitude, like '3 steps east'. In the line equation $\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}$, $\mathbf{a}$ is a position vector locating the line in space, while $\mathbf{d}$ is a direction vector defining its orientation.

First, check if their direction vectors are parallel (scalar multiples of each other). If they are not, try to find a point of intersection by setting their vector equations equal. This gives you three simultaneous equations for two variables ($\lambda$ and $\mu$). If you can't find a consistent solution for $\lambda$ and $\mu$ that works for all three equations, the lines do not intersect. Since they are not parallel and do not intersect, they must be skew.

Yes, absolutely. The formulas are just simpler. For 2D vectors $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j}$ and $\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j}$, the scalar product is $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2$. The geometric definition $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta$ and all its applications still hold.