Question 1

What's the difference between a position vector and a direction vector?

Accepted Answer

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.

Question 2

How do I know if lines are skew?

Accepted Answer

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.

Question 3

Can I use the dot product for 2D vectors?

Accepted Answer

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 $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos	heta$ and all its applications still hold.

Vectors

What you need to know

Vectors: Direction and Magnitude