2D and 3D Lines

The topics of 2D & 3D lines, the way the IB does them… I'm not a big fan.

### Representation of a Line

The general form of a line can be represented by the form:

(1)
\begin{align} \mathbf{r}=\mathbf{a}+t\mathbf{b} \end{align}

Where r, a and b are all vectors. a can be any point (in 2D or 3D) on the line and b can be thought of "directions" to another point on the line. The variable t basically tells us how many times to repeat the directions (once, twice, half, etc.).

### Direction Vector as Velocity

If the line being described is the trajectory of an object (and not just a geometric line) then the "direction" vector takes on additional meaning as a velocity vector.

Speed is the magnitude of the velocity vector.

(3)
$$Speed = |Velocity|$$

The IB folk often ask for the "speed." You need to recognize they are not asking for you to parrot back the velocity vector, but the magnitude of the velocity vector. This usually is a few easy points on a vector question.

The tough bits come when we look at parallel lines, perpendicular lines and intersection points...