Lesson 9 of 9 · 12 min

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Must-know facts

14 facts

1Direction cosines l, m, n are the cosines of the angles with the positive axes; l² + m² + n² = 1.
2Direction ratios are any three numbers proportional to l, m, n.
3l = ±a/√(a² + b² + c²), and likewise for m and n, with one sign throughout.
4Through P and Q: ratios x₂ − x₁, y₂ − y₁, z₂ − z₁; cosines are these over PQ.
5A, B, C are collinear when AB and BC have proportional ratios.
6Line: r = a + λb, or (x − x₁)/a = (y − y₁)/b = (z − z₁)/c.
7Line through two points a and b: r = a + λ(b − a).
8Acute angle between lines: cos θ = |b₁·b₂|/(|b₁||b₂|).
9Perpendicular: a₁a₂ + b₁b₂ + c₁c₂ = 0. Parallel: a₁/a₂ = b₁/b₂ = c₁/c₂.
10Skew lines are neither parallel nor meeting; their shortest segment is perpendicular to both.
11Skew distance: |(b₁ × b₂)·(a₂ − a₁)|/|b₁ × b₂|.
12Parallel distance: |b × (a₂ − a₁)|/|b|.
13Foot of perpendicular from S: λ = (s − a)·b/|b|²; image S′ = 2N − S (JEE).
14Plane: r·n = a·n or Ax + By + Cz = D; point distance |Ax₁ + By₁ + Cz₁ − D|/√(A² + B² + C²) (JEE).

Common traps

Where marks are lost

Squaring and adding direction ratios and expecting 1.

Only direction cosines square-sum to 1; divide the ratios by √(a² + b² + c²) first.

Reading ratios straight off (1 − x)/3 = (2y − 4)/5.

Rewrite as (x − 1)/(−3) = (y − 2)/(5/2) first; the ratios are −3, 5/2.

Leaving cos θ negative for the angle between two lines.

The angle between lines is taken acute: use |b₁·b₂|.

Using the skew-line formula for parallel lines.

For parallel lines b₁ × b₂ = 0; use |b × (a₂ − a₁)|/|b|.

Dropping the modulus in the shortest distance and reporting a negative length.

Distance is |(b₁ × b₂)·(a₂ − a₁)|/|b₁ × b₂|, never negative.

Calling two non-parallel lines in space intersecting by default.

In space they may be skew; check whether the shortest distance is 0.

Using the angle with the normal as the angle between a line and a plane.

They are complementary: sin φ = |b·n|/(|b||n|) (JEE).

Formulas

8 to know

Direction cosines from ratios

l = a/√(a² + b² + c²), m = b/√(…), n = c/√(…)

Take ± for the two senses of the line.

Through two points

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

PQ is the distance between the points.

Line (vector)

r = a + λb

a a point on the line, b its direction.

Line (Cartesian)

(x − x₁)/a = (y − y₁)/b = (z − z₁)/c

a, b, c direction ratios.

Angle between lines

cos θ = |b₁·b₂| / (|b₁| |b₂|)

Acute angle.

Skew lines

d = |(b₁ × b₂)·(a₂ − a₁)| / |b₁ × b₂|

0 means the lines meet.

Parallel lines

d = |b × (a₂ − a₁)| / |b|

Same direction b.

Point to plane (JEE)

|Ax₁ + By₁ + Cz₁ − D| / √(A² + B² + C²)

Plane Ax + By + Cz = D.

Key terms

6 terms

Direction cosines: cos α, cos β, cos γ for the angles a directed line makes with the positive axes.
Direction ratios: Any three numbers proportional to a line's direction cosines.
Skew lines: Lines in space that are neither parallel nor intersecting; no plane contains both.
Shortest distance: The length of the shortest segment joining two lines; for skew lines it is perpendicular to both.
Parameter: The number λ in r = a + λb; each value gives one point of the line.
Normal to a plane: A direction perpendicular to every line in the plane (JEE).