IB Maths AA HL Topic 3 — Geometry & Trigonometry Paper 1 & 2 ~6 min read HL only

Equation of a Line in Cartesian Form

The third way to write a line — eliminate λ from the parametric equations and stack them as a single chain of equal expressions: x − x₀l = y − y₀m = z − z₀n. Same line, no parameter.

📘 What you need to know

Cartesian form: x − x₀l = y − y₀m = z − z₀n (formula booklet).
(x₀, y₀, z₀) is a point on the line; (l, m, n) is a direction vector.
Numerators = (variable − point coordinate); denominators = direction components.
Read the components straight off — same point and direction as in vector/parametric forms.
If a direction component is zero: don’t divide by zero — write that variable as a constant separately.
Two zeros → two constants; write the one non-zero ratio equal to λ.
Point on line check: substitute coordinates; all three fractions must give the same value.

The Cartesian form

Cartesian equation of a line x − x₀l = y − y₀m = z − z₀n

It comes from making λ the subject of each parametric equation and setting them all equal. Each fraction equals the same λ for any point on the line.

Converting between forms

Parametric → Cartesian

make λ the subject
set all three equal

isolate λ in each, then chain them with =

Cartesian → vector

set each ratio = λ
solve for x, y, z

recover the parametric equations, then stack into r = a + λb

What if a direction component is zero?

You can’t divide by zero — so if l = 0, m = 0, or n = 0, that variable doesn’t change along the line. Write it separately as a constant.

If m = 0 (one zero) y = y₀, x − x₀l = z − z₀n

If l = 0 and n = 0 (two zeros) x = x₀, z = z₀, y − y₀m = λ

🧭 Recipe — convert vector form to Cartesian form

Read off the point and direction from r = a + λb: anchor (x₀, y₀, z₀) and direction (l, m, n).
Build three fractions: numerators are (x − x₀), (y − y₀), (z − z₀); denominators are l, m, n.
Chain them with equals signs.
Check for zero denominators — if any direction component is 0, write that variable as a constant separately.
Sanity check: pick a value of λ, find the point, plug into the Cartesian form. All three fractions should give the same number.

Worked examples

WE 1

Convert from vector form to Cartesian form

The line l has vector equation r = (3, −2, 5) + λ(2, 1, −3). Find the Cartesian equation of l.

Read off: point (3, −2, 5), direction (2, 1, −3) Apply the formula directly x − 32 = y + 21 = z − 5−3 numerators are (x − x₀); a “+2” came from y − (−2). The middle fraction can also be written as just (y + 2).

WE 2

Convert from parametric form to Cartesian form

A line has parametric equations x = 4 + 2λ, y = −1 − 5λ, z = 3 + λ. Find the Cartesian equation of the line.

Make λ the subject in each equation x = 4 + 2λ → λ = (x − 4)/2 y = −1 − 5λ → λ = (y + 1)/(−5) z = 3 + λ → λ = z − 3 Set all three expressions for λ equal x − 42 = y + 1−5 = z − 3 when the direction component is 1, the denominator is invisible — z − 3 alone

WE 3

Convert from Cartesian form to vector form

A line has Cartesian equation x − 23 = y + 4−1 = z − 62. Find the vector equation of the line.

Step 1: Set each ratio equal to λ (x − 2)/3 = λ → x = 2 + 3λ (y + 4)/−1 = λ → y = −4 − λ (z − 6)/2 = λ → z = 6 + 2λ Step 2: Stack into vector form r = (2, −4, 6) + λ(3, −1, 2) point comes from numerators; direction is the denominators

WE 4

Cartesian equation through two points

Find the Cartesian equation of the line passing through A(1, 3, −2) and B(7, −1, 4).

Step 1: Direction AB = B − A AB = (7−1, −1−3, 4−(−2)) = (6, −4, 6) Step 2: Simplify by 2 → (3, −2, 3) Step 3: Use A as the anchor point x − 13 = y − 3−2 = z + 23 check B: (7−1)/3 = 2; (−1−3)/−2 = 2; (4+2)/3 = 2 ✓

WE 5

Cartesian form with a zero direction component

Find the Cartesian equation of the line passing through (−3, 2, 5) with direction vector 4i − 2k.

Step 1: Direction = (4, 0, −2) — middle component is zero Step 2: y stays constant, equal to its starting value y = 2 Step 3: Write the remaining ratio for x and z y = 2, x + 34 = z − 5−2 never write a fraction with 0 in the denominator — split that variable off as a constant

WE 6

Check if a point lies on a line in Cartesian form

Determine whether the point P(7, −6, 13) lies on the line with Cartesian equation x − 12 = y + 3−1 = z − 43.

Substitute the coordinates into each fraction (7 − 1)/2 = 6/2 = 3 (−6 + 3)/−1 = −3/−1 = 3 ✓ (13 − 4)/3 = 9/3 = 3 ✓ P lies on the line (all give 3) the common value 3 is the parameter λ at this point

💡 Top tips

The numerator is (variable − point coordinate); the denominator is the direction component.
Check signs carefully: y − (−2) becomes y + 2 in the numerator.
Zero in a direction component → write that variable as a constant; don’t put 0 in a denominator.
Each fraction equals λ — that’s why you can chain them with =.
Same value across all fractions means the point lies on the line.

⚠ Common mistakes

Dividing by zero when a direction component is zero — write that variable as a separate constant equation.
Sign mix-up in numerators: x − (−3) is x + 3, not x − 3.
Reading point and direction in the wrong slots — point goes in numerators, direction in denominators.
Concluding “point on line” too early after one fraction matches — must check all three.
Forgetting y₀ = 0 when one parametric equation has no constant: y = 5λ means y₀ = 0, so the numerator is just y.

Next: Applications to Kinematics. The vector equation of a line is also the equation of motion for an object moving with constant velocity: r = r₀ + vt. Same algebra, different story — position, velocity, and time replacing anchor, direction, and parameter.

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