Direction cosines from ratios
Three Dimensional Geometry · Direction Cosines and Direction Ratios\[l=\frac{a}{\sqrt{a^2+b^2+c^2}},\ m=\frac{b}{\sqrt{a^2+b^2+c^2}},\ n=\frac{c}{\sqrt{a^2+b^2+c^2}}\]
Chapter 11 · Vectors and Three-Dimensional Geometry
Three Dimensional Geometry
Chapter Overview
In three dimensions, a line is fixed by a point on it and its direction. This chapter describes directions with direction cosines and direction ratios, writes the equation of a line in vector and Cartesian form, finds the angle between two lines, and computes the shortest distance between two skew lines – lines that are neither parallel nor intersecting.
Board focus: A long answer on shortest distance or on the foot of a perpendicular is very common; MCQs test direction cosines and angles.
Topics
Key Concepts
- Direction cosines \(l,m,n\): \(l^2+m^2+n^2=1\); direction ratios \(a,b,c\) are proportional to them.
- Line through \(\vec a\) parallel to \(\vec b\): \(\vec r=\vec a+\lambda\vec b\); Cartesian \(\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\).
- Angle between lines: \(\cos\theta=\frac{|\vec b_1\cdot\vec b_2|}{|\vec b_1||\vec b_2|}\).
- Shortest distance between skew lines: \(d=\frac{|(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)|}{|\vec b_1\times\vec b_2|}\).
Formulas
Equation of a line
Three Dimensional Geometry · Equation of a Line in Space\[\vec r=\vec a+\lambda\vec b,\qquad \frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\]
Angle between lines
Three Dimensional Geometry · Angle Between Two Lines\[\cos\theta=\frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{\sum a_1^2}\sqrt{\sum a_2^2}}\]
Shortest distance
Three Dimensional Geometry · Shortest Distance Between Two Lines\[d=\frac{|(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)|}{|\vec b_1\times\vec b_2|}\]
Key Points
General point on a line
Three Dimensional Geometry · Equation of a Line in SpaceOn \(\frac{x-1}{2}=\frac{y+3}{-1}=\frac{z}{4}=\lambda\), every point is \((1+2\lambda,\ -3-\lambda,\ 4\lambda)\).
Intersecting lines
Three Dimensional Geometry · Shortest Distance Between Two LinesTwo non-parallel lines intersect exactly when their shortest distance is 0 (they are coplanar).
Common Mistakes
DRs are not DCs
Three Dimensional Geometry · Direction Cosines and Direction Ratios\(2,-1,2\) are direction ratios; the direction cosines are \(\frac23,-\frac13,\frac23\).
Reading DRs from non-standard forms
Three Dimensional Geometry · Equation of a Line in SpaceIn \(\frac{1-x}{2}=\frac{y}{3}=\frac{2z+1}{4}\), rewrite as \(\frac{x-1}{-2}=\frac{y}{3}=\frac{z+\frac12}{2}\): DRs are \(-2,3,2\).
Solved Examples
Shortest distance
Three Dimensional Geometry · Shortest Distance Between Two Lines\(\vec r=(\hat i+\hat k)+\lambda(\hat i+\hat j)\) and \(\vec r=(2\hat j-\hat k)+\mu(\hat j+\hat k)\).
\(\vec b_1\times\vec b_2=\hat i-\hat j+\hat k\), \(\vec a_2-\vec a_1=-\hat i+2\hat j-2\hat k\); \(d=\frac{|-1-2-2|}{\sqrt3}=\frac{5}{\sqrt3}\).
Practice & Tests
Chapter Practice
Untimed mixed questions from all topics. Answers are revealed after you submit.
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