Revise Vector equations of lines for IB Mathematics: Analysis and Approaches (3.8)

To fix a line in space you need exactly two things: somewhere it passes through, and which way it points. The first is a point, captured by its position vector; the second is a direction, captured by a direction vector. Everything else on the line is reached by starting at the point and travelling some multiple of the direction.

$\mathbf{r}=\mathbf{a}+\lambda\mathbf{b}$

Neither ingredient is unique. Any point on the line can serve as $\mathbf{a}$, and any scalar multiple of $\mathbf{b}$ — larger, smaller, or reversed — is an equally good direction. So the very same line has infinitely many correct vector equations, which is why markers award full credit to any valid anchor-and-direction pair, not just the 'textbook' one.

The vector equation is compact, but its component form is what you actually compute with. Reading off each coordinate line by line gives the parametric form; eliminating the parameter from those three equations gives the Cartesian form, which is convenient for finding where a line meets a plane or surface.

Vector form: $\begin{pmatrix} x \\ y \\ z \end{pmatrix}=\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}+\lambda\begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix}$

Parametric form: $x=a_1+\lambda b_1,\quad y=a_2+\lambda b_2,\quad z=a_3+\lambda b_3$

Cartesian form (each $b_i\neq0$): $\dfrac{x-a_1}{b_1}=\dfrac{y-a_2}{b_2}=\dfrac{z-a_3}{b_3}\;(=\lambda)$

Given two points, the direction is the vector from one to the other, $\vec{AB}=\vec{OB}-\vec{OA}$ — a subtraction, never an addition. Anchor at either point and you are done. To test whether some other point lies on the line, substitute it for $\mathbf{r}$ and demand a single value of $\lambda$ that satisfies every component at once: one stride length must set all three coordinates together.

The angle between two lines is the angle between their directions — where the lines actually sit in space is irrelevant, and they need not even meet. Use the scalar (dot) product of the direction vectors. Because a line points 'both ways', we always report the acute angle: if the cosine comes out negative, take the absolute value before applying $\arccos$.

$\cos\theta=\dfrac{\mathbf{b}_1\cdot\mathbf{b}_2}{|\mathbf{b}_1|\,|\mathbf{b}_2|}$, and the angle between the lines is $\theta=\arccos\left(\dfrac{|\mathbf{b}_1\cdot\mathbf{b}_2|}{|\mathbf{b}_1|\,|\mathbf{b}_2|}\right)$ (the acute angle).

In three dimensions two lines fall into exactly one of three cases. They may be parallel (same direction, up to scaling), they may intersect at a single point, or they may be skew — non-parallel yet never meeting, like two aircraft on crossing headings but at different altitudes. Skewness is genuinely new in 3D; it cannot happen in a plane.

The method is always the same. First compare directions: proportional means parallel, and you are done. If not parallel, set the two equations equal, giving three component equations in the two unknowns $\lambda$ and $\mu$ — use different letters for the two parameters, or the whole test collapses. Solve two of the equations, then substitute into the third: consistent means the lines intersect, and back-substitution gives the point; inconsistent means the lines are skew.

On Paper 1 the marks are analytic: each is tied to a specific line of working — a method mark ($M$) for the correct approach, or an accuracy mark ($A$) for the right value, where an $A$ mark is dependent on the $M$ mark before it. Follow-through (FT) means that a right method carried out on a wrong earlier number still earns the marks that depend on it. But this protection only exists if the method is on the page. Study how each of the five marks below is earned by a specific line.