����������� � ����������THREE DIMENSIONAL GEOMETRY� By� Gopinathan M V� PGT Mathematics� JNV Sindhudurg� Maharashtra� Pune Region ��

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Suppose a line passing through origin is making angles α,β,γ with x-axis,y-axis,z-axis respectively then α,β,γ are called direction angles, then cosine of these angles cosα,cosβ,cosγ are called direction cosines of the directed line l.

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β

γ

α

O

z

l

y

• If we reverse the direction of l then direction angles are replaced by their supplements' i.e. π-α,π-β,π-γ.Then the sign of direction cosines are reversed.

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• In order to have a unique set of direction cosines ,we must take given line as directed line. these unique direction cosines are denoted by l,m,n.

• If the given line in space does not passes through the origin, then in order to find it’s direction cosines we draw a line through the origin and parallel to the given line.

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• Any three numbers which are proportional to the direction cosines are called direction ratios and are denoted by a,b,c.

Note:

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here , l=ak , m= bk , n=ck

therefore if a,b,c are direction ratios of a linethen its direction cosines of the line are

l=

b

m=

l

n=

• For any line if a,b,c are direction ratios then ka,kb,kc[k≠0] also can be taken as direction ratios.

Note:if l,m,n are direction cosines of a line then l² +m²+n²=1

Note:direction cosines of aline passing through two points(x,y,z)and (x₂,y₂,z₂) is given by

x₂-x₁, y₂-y₁, z₂-z₁

Note: Direction cosines of X axis are 1,0,0

Direction cosines of Y axis are 0,1,0

Direction cosines of Z axis are 0,0,1

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Equation of a line in Space

Angle Between Two lines

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Note:- If a₁,b₁,c₁, and a₂,b₂,c₂, are the direction ratios of two lines then

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i) the lines are parallel iff

ii) the lines are perpendicular iff a₁a₂+b₁b₂+c₁c₂=0

Equation of a Plane

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EQUATION OF PLANE THROUGH THE INTERSECTION OF TWO PLANES

Consider two planes new plane

a₁x+b₁y+c₁z+d₁=0 1

a₂x+b₂y+c₂z+d₂=0 2

The equation of plane through the

intersection of the two plane is given

By: 1+λ2=0 where λ is a scalar

(a₁x+b₁y+c₁z+d₁)+ λ(a₂x+b₂y+c₂z+d₂)=0

1

2

Questions!

1)find the equalation of the plane passing through intersection of the planes.

r (2i+2j-3k)=7 r (2i+5j+3k)=9

And through the point (2,1,3)

2)Find the equation of the plane through the line of

intersection of the planes.

x+y+2=1 and 2x+3y+42=5 which is perpendicular to the plane x-y+z=0 .

FINDING EQUATION OF THE PLANE PASSING THROUGH ONE POINT AND SATISFYING TWO MORE CONDITIONS

Note:equation of any plane passing through (x₁,y₁,z₁)

can be taken as A(x-x₁)+B(y-y₁)+C(z-z₁)=0 where A,B,C

are direction ratios of normal to the plane

normal

A,B,C

(x₁,y₁,z)

QUESTION

IMPORTANT TYPE OF PROBLEMS

1)Finding the equation of plane passing through three noncollinear points.

2)verifying four points are coplanar and finding the equation of plane containing these points.

3)Finding image of a point with respect to a plane/line.

4)Finding the foot of a perpendicular from a point to a plane/line.

5)Finding the distance of a point from plane measured parallel to a line.

6)Finding the distance of point from a line measured parallel to a plane.