Equations of Lines

At the boards today, the challenge was to describe a straight path so precisely that another group could walk it — using only vector language. Every group converged on the same two ingredients: a point to start from, and a direction to keep. That pair is a line, and writing it down gives the vector equation:

$\vec{r}={\vec{r}}_0+t\vec{m}$

Start at the point ${\vec{r}}_0$, walk multiples $t$ of the direction vector $\vec{m}$. Every value of the parameter $t$ lands on the line; negative $t$ walks backward; $t$ sweeps through all real numbers and the whole line appears. It is adding and scaling doing geometry.

Three costumes for one line

Take the line through $(1,2)$ with direction $(3,1)$. It owns three equivalent descriptions:

Form	The line through $(1,2)$, direction $(3,1)$
Vector	$\vec{r}=(1,2)+t(3,1)$
Parametric	$x=1+3t,y=2+t$
Scalar	$x-3y+5=0$

The parametric form is the vector form read one coordinate at a time. The scalar form comes from eliminating $t$ — solve each parametric equation for $t$, set the results equal, tidy up. Each costume answers a different question fastest: the vector form knows direction, the parametric form generates points on demand, and the scalar form tests a point’s membership instantly. Moving fluently among them is the skill; none of them is “the” equation.

The surprise in three-space

Lift everything into three dimensions and the vector and parametric forms come along without complaint — the line through $(3,2,-1)$ and $(0,2,1)$ has direction $(-3,0,2)$ and equation $\vec{r}=(3,2,-1)+t(-3,0,2)$. But the scalar form does not survive the trip. A single scalar equation in $x$, $y$, $z$ has too many solutions to be a line — its solution set is a whole plane — so no single scalar equation can pin down a line in three-space. The best scalar language can do is name a line as the meeting of two planes: two equations, jointly satisfied. Check it against the crease where two walls of the room meet — each wall is one equation, the crease needs both.

That asymmetry — vector forms generalise, scalar forms buckle — is the reason vector equations run the rest of the unit. Explore it with your own hands in Using Desmos before trusting it, sketch your findings, and then take on Lines and Planes Practice. The The Flight Path task starts here: an approach path is nothing but a point and a direction.

Curriculum connection

C3.1

recognize that the solution points $(x,y)$ in two-space of a single linear equation in two variables form a line and that the solution points $(x,y)$ in two-space of a system of two linear equations in two variables determine the point of intersection of two lines, if the lines are not coincident or parallel Sample problem: Describe algebraically the situations in two-space in which the solution points $(x,y)$ of a system of two linear equations in two variables do not determine a point.

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C4.1

recognize a scalar equation for a line in two-space to be an equation of the form $Ax+By+C=0$, represent a line in two-space using a vector equation (i.e., $\vec{r}={\vec{r}}_0+t\vec{m}$) and parametric equations, and make connections between a scalar equation, a vector equation, and parametric equations of a line in two-space

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C4.2

recognize that a line in three-space cannot be represented by a scalar equation, and represent a line in three-space using the scalar equations of two intersecting planes and using vector and parametric equations (e.g., given a direction vector and a point on the line, or given two points on the line) Sample problem: Represent the line passing through $(3,2,-1)$ and $(0,2,1)$ with the scalar equations of two intersecting planes, with a vector equation, and with parametric equations.

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