Lesson 9 of 9 · 12 min
Chapter review
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Must-know facts
14 facts
- 01Direction cosines l, m, n are the cosines of the angles with the positive axes; l² + m² + n² = 1.
- 02Direction ratios are any three numbers proportional to l, m, n.
- 03l = ±a/√(a² + b² + c²), and likewise for m and n, with one sign throughout.
- 04Through P and Q: ratios x₂ − x₁, y₂ − y₁, z₂ − z₁; cosines are these over PQ.
- 05A, B, C are collinear when AB and BC have proportional ratios.
- 06Line: r = a + λb, or (x − x₁)/a = (y − y₁)/b = (z − z₁)/c.
- 07Line through two points a and b: r = a + λ(b − a).
- 08Acute angle between lines: cos θ = |b₁·b₂|/(|b₁||b₂|).
- 09Perpendicular: 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
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).