10 min readConcept sheet1-mark MCQ, 2-mark question; used inside 5-mark questions

Cartesian equation of a line

A line through (x₁, y₁, z₁) with direction ratios a, b, c is (x − x₁)/a = (y − y₁)/b = (z − z₁)/c. Each ratio equals λ.

Cartesian form(x − x₁)/a = (y − y₁)/b = (z − z₁)/c

With direction cosines(x − x₁)/l = (y − y₁)/m = (z − z₁)/n

Through two points(x − x₁)/(x₂ − x₁) = (y − y₁)/(y₂ − y₁) = (z − z₁)/(z₂ − z₁)

General point(x₁ + aλ, y₁ + bλ, z₁ + cλ)

From the vector form to the Cartesian form

Start with r⃗ = a⃗ + λb⃗, where a⃗ = x₁î + y₁ĵ + z₁k̂ and b⃗ = aî + bĵ + ck̂.

1
Write r⃗ = xî + yĵ + zk̂ for any point (x, y, z) on the line.
2
xî + yĵ + zk̂ = (x₁ + λa)î + (y₁ + λb)ĵ + (z₁ + λc)k̂
3
Equate components: x = x₁ + λa, y = y₁ + λb, z = z₁ + λc.
4
Make λ the subject of each: λ = (x − x₁)/a, λ = (y − y₁)/b, λ = (z − z₁)/c.
5★
So, (x − x₁)/a = (y − y₁)/b = (z − z₁)/c is the Cartesian equation.

★ key step

Reading a line in standard form

In standard form, x, y and z each have coefficient +1 in the numerators.
(1 − x)/2 means (x − 1)/(−2), so the DR is −2, not 2.
(2z − 1)/6 means 2(z − 1/2)/6 = (z − 1/2)/3, so the DR is 3 and the point has z = 1/2.
A zero denominator, as in (y − 3)/0, means y = 3 all along the line: the line is perpendicular to the y-axis.
x/4 means (x − 0)/4, so the point has x = 0.

Example 1: parallel to a given line

Problem

Find the Cartesian and vector equations of the line through (3, −2, 5) parallel to the line (x − 1)/2 = (y + 4)/(−3) = z/6.

Working

DRs of the given line: 2, −3, 6. A parallel line has the same DRs.
Cartesian: (x − 3)/2 = (y + 2)/(−3) = (z − 5)/6
Vector: a⃗ = 3î − 2ĵ + 5k̂, b⃗ = 2î − 3ĵ + 6k̂
r⃗ = (3î − 2ĵ + 5k̂) + λ(2î − 3ĵ + 6k̂)

Answer

(x − 3)/2 = (y + 2)/(−3) = (z − 5)/6; r⃗ = (3î − 2ĵ + 5k̂) + λ(2î − 3ĵ + 6k̂).

Example 2: a line not in standard form

Problem

For the line (4 − x)/2 = (y + 1)/3 = (3 − 2z)/4, find a point on it, its direction cosines and its vector equation.

Working

(4 − x)/2 = (x − 4)/(−2)
(3 − 2z)/4 = −2(z − 3/2)/4 = (z − 3/2)/(−2)
Standard form: (x − 4)/(−2) = (y + 1)/3 = (z − 3/2)/(−2)
Point: (4, −1, 3/2). DRs: −2, 3, −2.
√(4 + 9 + 4) = √17, so the DCs are −2/√17, 3/√17, −2/√17.
Vector form: r⃗ = (4î − ĵ + (3/2)k̂) + λ(−2î + 3ĵ − 2k̂)

Answer

Point (4, −1, 3/2); DCs −2/√17, 3/√17, −2/√17; r⃗ = (4î − ĵ + (3/2)k̂) + λ(−2î + 3ĵ − 2k̂).

Example 3: through two points

Problem

Find the Cartesian equation of the line through A(2, −1, 3) and B(4, 0, 5). Find k if C(k, 1, 7) lies on it.

Working

DRs of AB: 4 − 2, 0 − (−1), 5 − 3 = 2, 1, 2
Line: (x − 2)/2 = (y + 1)/1 = (z − 3)/2 = λ
General point: (2 + 2λ, −1 + λ, 3 + 2λ)
For C, y = 1: −1 + λ = 1, so λ = 2.
z = 3 + 2(2) = 7, which matches C.
x = 2 + 2(2) = 6, so k = 6.

Answer

(x − 2)/2 = (y + 1)/1 = (z − 3)/2, and k = 6.

Vector form vs Cartesian form

Vector formr⃗ = a⃗ + λb⃗. Uses the position vector of a point and a direction vector. Handy with dot and cross products.

Cartesian form(x − x₁)/a = (y − y₁)/b = (z − z₁)/c. Uses the coordinates of a point and the direction ratios. Handy for writing a general point.

Both describe the same line. Be ready to convert either way.

In real life

3D printers and design software store a straight edge as a point plus direction ratios, which is exactly the Cartesian equation of a line.

Exam angle

1 marks: “Write the direction ratios of the line (x − 2)/3 = (1 − y)/4 = (z + 5)/2.” (1 − y)/4 = (y − 1)/(−4). So the DRs are 3, −4, 2.

2 marks: “Find the Cartesian and vector equations of the line through (1, −1, 2) parallel to 2î + 3ĵ − k̂.” (x − 1)/2 = (y + 1)/3 = (z − 2)/(−1); r⃗ = (î − ĵ + 2k̂) + λ(2î + 3ĵ − k̂).

2 marks: “The Cartesian equation of a line is 2x − 4 = 3y + 3 = 6 − 6z. Find its direction ratios and its vector equation.” Write 2(x − 2) = 3(y + 1) = −6(z − 1) and divide by 6: (x − 2)/3 = (y + 1)/2 = (z − 1)/(−1). DRs 3, 2, −1; r⃗ = (2î − ĵ + k̂) + λ(3î + 2ĵ − k̂).

Remember it

The coefficient of x, y and z must be +1 before you read the point and the DRs.