Lines & Planes · Grades 12–12

Lines & Planes Worksheets

Grade 12 worksheets on lines and planes in 2-space and 3-space: writing a line's vector and parametric equations from a point and direction (or two points), reading a plane's normal vector from its scalar equation, building a plane's equation from a normal and a point (or from three points), and finding the distance from a point to a plane.

Generate lines & planes worksheets →

Free plan · 5 credits every month · answer keys included · no card required.

What it is

A line's vector equation is r = r₀ + t·m: start at a known point r₀ and travel any distance t along a direction vector m. A plane's scalar equation Ax+By+Cz=D has normal vector ⟨A, B, C⟩ — perpendicular to every direction lying in the plane. Given a normal and a point, or three points, that scalar equation can always be rebuilt.

Why it matters

This is the algebra behind 3D modelling, computer graphics, and physics simulations, anywhere a computer needs to know exactly where a flat surface or a straight path sits in space.

How to do it

  1. Vector equation of a line: r = r₀ + t·m, where r₀ is a point on the line and m is a direction vector.
  2. Parametric equations: split the vector equation into one line per coordinate — x = x₀ + t·mx, y = y₀ + t·my (and z = z₀ + t·mz in 3-space).
  3. Two points on a line: the direction vector is (second point) minus (first point).
  4. A plane's normal vector is read straight off its scalar equation: Ax+By+Cz=D has normal ⟨A, B, C⟩.
  5. Plane through three points: build two direction vectors from the points, cross them for a normal, then use point-normal form.
  6. Point-to-plane distance: d = |Ax₀+By₀+Cz₀−D| / √(A²+B²+C²), using the point's coordinates and the plane's coefficients.

Examples by level

  • BeginnerA line passes through P(-6, 6) with direction vector m⃗=⟨3,0⟩\vec{m} = \langle 3, 0 \rangle. Write its vector equation =r⃗=⟨−6,6⟩+t⟨3,0⟩\vec{r} = \langle -6, 6 \rangle + t\langle 3, 0 \rangle
  • IntermediateA line passes through P(2, -3, -3) with direction vector m⃗=⟨−5,0,8⟩\vec{m} = \langle -5, 0, 8 \rangle. Write its vector equation =r⃗=⟨2,−3,−3⟩+t⟨−5,0,8⟩\vec{r} = \langle 2, -3, -3 \rangle + t\langle -5, 0, 8 \rangle
  • AdvancedA line passes through P(2, 4) with direction vector m⃗=⟨0,−7⟩\vec{m} = \langle 0, -7 \rangle. Write its parametric equations =x = 2 + 0t, y = 4 + -7t

Examples are generated by the same engine as the worksheets, so they're always mathematically correct. Built to avoid repeats, not reshuffle the same handful of questions.

Common mistakes

  • Trying to write a 3-space line with a single scalar equation — that only works in 2-space; 3-space lines need a vector or parametric equation.
  • Reading the normal vector's components in the wrong order, or forgetting the constant on the right side isn't part of the normal.
  • Using the wrong point when converting from two points to a direction vector (direction is always 'second point minus first point').
  • Forgetting the absolute value in the point-to-plane distance formula.

Tips

  • A 3-space line can never be written as one scalar equation — that only works for a plane in 3-space. Use the vector or parametric form instead.
  • Direction vector between two points is always 'second point minus first point', not the other way around.
  • A plane's normal vector components are exactly its equation's x, y, z coefficients — nothing to compute.
  • The point-to-plane distance formula always uses the absolute value of the numerator — a distance can't come out negative.

For parents

A line's vector equation is just 'start point plus however far you've travelled in that direction' — r = r₀ + t·m is that sentence written in symbols, with t as the 'how far' dial.

For teachers

Lead with the fact a 3-space line can't be written as a single equation the way a 2-space line can — that's the concept students most often try to force, and naming it up front heads off a lot of confusion before the vector/parametric forms are introduced.

Key vocabulary

direction vector
a vector showing which way a line travels; any nonzero scalar multiple of it works just as well
vector equation
r = r₀ + t·m — a line written as a starting point plus a direction, scaled by a parameter t
normal vector
a vector perpendicular to every direction lying in a plane
parametric equations
a line's vector equation split into one equation per coordinate (x=…, y=…, z=…), each in terms of t
scalar equation
a plane written as Ax+By+Cz=D — one linear equation in three variables

Frequently asked questions

Why can't a 3-space line have a scalar equation like a 2-space line does?
A single linear equation in three variables (Ax+By+Cz=D) describes a whole plane, not a line — a 3-space line needs a vector or parametric equation instead.
How do you find a plane's normal vector?
If the plane's scalar equation is Ax+By+Cz=D, the normal vector is simply ⟨A, B, C⟩ — no calculation needed.
How do you find the equation of a plane through three points?
Build two vectors from the points, take their cross product to get a normal vector, then use that normal and any one of the three points to write the scalar equation.
How do you find the distance from a point to a plane?
d = |Ax₀ + By₀ + Cz₀ − D| / √(A² + B² + C²), using the plane's coefficients A, B, C, D and the point's coordinates (x₀, y₀, z₀). The absolute value keeps the distance from coming out negative.

Ready to practice?

Generate a print-ready lines & planes pack with answer keys: pick a grade and difficulty, and it's ready in seconds.

Create free worksheets

Free to start · answer keys included · your first pack unlocks 14 days of Standard · no card required.