Three Dimensional Geometry

Board: CBSE |Class:Class12

ComprehensivestudynotesforThreeDimensionalGeometrybyAjayYadav.

KeyConcepts

DirectionCosinesandRatios

DC:l=cosα,m=cosβ,n=cosγ.l²+m²+n²=1.DR:a,b,cproportionaltol,m,n.Directionratiosoflinejoining(x₁,y₁,z₁)and(x₂,y₂,z₂):(x₂-x₁,y₂-y₁,z₂-z₁).

EquationofaLine

Vectorform:r=a+λb.Cartesian:(x-x₁)/a=(y-y₁)/b=(z-z₁)/c.Two-point:(x-x₁)/(x₂-x₁)=(y-y₁)/(y₂-y₁)=(z-z₁)/(z₂-z₁).

AngleBetweenLines

cosθ=|a₁a₂+b₁b₂+c₁c₂|/√(Σa₁²+b₁²+c₁²).Linesareperpendicularifa₁a₂+b₁b₂+c₁c₂=0.

ShortestDistance

Betweenskewlines:SD=|(a₂-a₁)·(b₁×b₂)|/|b₁×b₂|. Between parallel lines: SD = |(a₂-a₁)×b|/|b|.

EquationofaPlane

Normal:r&middotn=d(orax+by+cz+d=0).Intercept:x/p+y/q+z/r=1.Anglebetweenplanes:cosθ=|a₁a₂+b₁b₂+c₁c₂|/√(Σa₁²).

DistancefromPlane

Point(x₁,y₁,z₁)toplaneax+by+cz+d=0:d=|ax₁+by₁+cz₁+d|/√(a²+b²+c²).

ImportantFormulas

DCproperty	l²+m²+n²=1
Lineequation	(x-x₁)/a=(y-y₁)/b=(z-z₁)/c
Plane(normal)	ax+by+cz+d=0
Point-planedistance	d=|ax1+by1+cz1+d|/√(a²+b²+c²)

SolvedExamples

Example1:FindDCoflinewithDR1,2,2.

Solution:√(1+4+4)=3.DC=(1/3,2/3,2/3).

Example2:Find anglebetweenlines(x-1)/2=(y+2)/3=(z-3)/4and(x-2)/5=(y-1)/6=(z+4)/7.

Solution:cosθ=|2(5)+3(6)+4(7)|/√(4+9+16)√(25+36+49)=|10+18+28|/√29√110=56/√3190.

Example3:Finddistanceof(1,2,3)fromplane2x+3y+4z-20=0.

Solution:d=|2+6+12-20|/√4+9+16 = 0/√29 = 0 (point lies on plane).

Practice Questions

1. Find DC of line (1,2,3) to (4,5,6).
2. Find equation of line through (1,2,3) with DR (2,-1,3).
3. Find angle between planes 2x+y+z=5 and x+2y+z=7.
4. Find SD between skew lines: (x-1)/2=(y-2)/3=(z-3)/4 and x/5=(y-1)/6=(z+2)/7.
5. Find distance of origin from plane 2x-3y+6z=21.

Download PDF

Download PDF notes.

Video Lessons

Watch on Videos page.