7 min readConcept sheet1-mark MCQ or 2-mark question; the first step of many 3 and 5-mark line questions

Direction cosines and ratios of a line joining two points

The line through P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) has direction ratios x₂ − x₁, y₂ − y₁, z₂ − z₁. Divide by PQ for the direction cosines.

DRs of PQx₂ − x₁, y₂ − y₁, z₂ − z₁

Length PQ√[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]

DCs of PQ(x₂ − x₁)/PQ, (y₂ − y₁)/PQ, (z₂ − z₁)/PQ

A, B, C are collinear whenThe DRs of AB and BC are proportional (B is common to both)

Why the differences are direction ratios

Let O be the origin, and let P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) be two points.

1
Position vectors: OP⃗ = x₁î + y₁ĵ + z₁k̂ and OQ⃗ = x₂î + y₂ĵ + z₂k̂.
2
PQ⃗ = OQ⃗ − OP⃗ = (x₂ − x₁)î + (y₂ − y₁)ĵ + (z₂ − z₁)k̂.
3
The components of a vector along a line are direction ratios of that line. So x₂ − x₁, y₂ − y₁, z₂ − z₁ are DRs of PQ.
4
|PQ⃗| = PQ = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²].
5
The unit vector along PQ is PQ⃗/PQ. Its components are the direction cosines.
6★
So, the direction cosines of PQ are (x₂ − x₁)/PQ, (y₂ − y₁)/PQ, (z₂ − z₁)/PQ.

★ key step

Example 1: direction cosines from two points

Problem

Find the direction ratios and direction cosines of the line joining P(1, −2, 3) and Q(3, 1, −3).

Working

DRs: 3 − 1, 1 − (−2), −3 − 3 = 2, 3, −6
PQ = √(2² + 3² + (−6)²) = √(4 + 9 + 36) = √49 = 7
DCs: 2/7, 3/7, −6/7
Check: 4/49 + 9/49 + 36/49 = 49/49 = 1

Answer

DRs 2, 3, −6; DCs 2/7, 3/7, −6/7.

Example 2: are the points collinear?

Problem

Show that A(1, 2, 3), B(3, 5, 4) and C(7, 11, 6) are collinear.

Working

DRs of AB: 3 − 1, 5 − 2, 4 − 3 = 2, 3, 1
DRs of BC: 7 − 3, 11 − 5, 6 − 4 = 4, 6, 2
4/2 = 6/3 = 2/1 = 2, so the DRs of BC are 2 times those of AB.
AB and BC are parallel and share the point B, so they lie on one line.

Answer

A, B and C are collinear.

Proportional direction ratios: parallel or collinear?

Parallel linesTwo lines with proportional DRs point the same way. They may still be different lines, like the two rails of a railway track.

Collinear pointsAB and BC have proportional DRs and share the point B. Only then do A, B and C lie on one line.

Always mention the common point when you prove collinearity.

Common slips

Subtract in the same order for all three coordinates: x₂ − x₁, y₂ − y₁, z₂ − z₁.
Watch the signs: 1 − (−2) = 3, not −1.
The DCs of QP are the negatives of the DCs of PQ.
If a difference is 0, that DC is 0: the line is perpendicular to that axis.

In real life

On a cable-stayed bridge like the Bandra-Worli Sea Link in Mumbai, each steel cable runs between two fixed points. Engineers find its direction from the coordinates of its two ends.

Exam angle

1 marks: “Write the direction ratios of the line joining (2, −1, 4) and (5, 3, 4).” 5 − 2, 3 − (−1), 4 − 4 = 3, 4, 0. The line is perpendicular to the z-axis.

2 marks: “Find the direction cosines of the line joining A(4, −3, 2) and B(2, 3, −1).” DRs: −2, 6, −3. AB = √(4 + 36 + 9) = 7. DCs: −2/7, 6/7, −3/7.

2 marks: “Show that the points A(0, 1, 2), B(2, 5, 8) and C(−1, −1, −1) are collinear.” AB has DRs 2, 4, 6 and AC has DRs −1, −2, −3. AC's DRs are −1/2 times AB's, and A is common. So the points are collinear.

Remember it

Second minus first gives the DRs; divide by the length for the DCs.