Three dimensional geometry

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notes of three dimensional geometry class XII

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Three dimensional geometry

  1. 1. CLASS XII THREE DIMENSIONAL GEOMETRY DIRECTION COSINES& DIRECTION RATIOS OF A LINE The direction cosines of a line are defined as the direction cosines of any vector whose support is a given line. If , , ,α β γ are the angles which the line l makes with the positive direction of x-axis, y-axis & z- axis respectively,then its direction cosines are cosα , cos β , cos γ . Or -cosα , -cos β , -cos γ . Therefore, if l, m, n are D.C of a line, then -l,-m,-n are also its D.C & we always have 222 nml ++ =1. DIRECTION RATIOS OF LINE:- Any three numbers which are proportional to the D.C of a line are called D .R of a line. If l ,m ,n are D.C and a, b ,c are D.R of a line then a=λ l, b λm, c=λn. TO FIND DIRCTION COSINES OF A LINE FROM ITS DIRECTION RATIO’S Let <a, b ,c> be the D.R of a line L and <l ,m ,n>be its D.C then a=λl, b=λ m, c=λn. For some λ (≠0)  l=a/λ, m=b/λ, n=c/λ As 222 nml ++ =1 => 2 2 2 2 2 2 1 a b c λ λ λ + + =  λ = 2 2 2 a b c± + +  2 2 2 a l a b c = ± + + , 2 2 2 b m a b c = ± + + , 2 2 2 c n a b c = ± + + DIRECTION RATIOS OF A LINE PASSING THROUGH TWO POINTS The D.C. of a line joining two points P( 1 1 1, ,x y z ) &Q ( 2 2 2, ,x y z ) are 2 1 2 1 2 1 , , x x y y Z Z PQ PQ PQ − − − < > Where PQ= 2 2 2 2 1 2 1 2 1( ) ( ) ( )x x y y z z− + − + − Direction Ratios of a line joining the points P( 1 1 1, ,x y z ) & Q( 2 2 2, ,x y z ) are 2 1 2 1 2 1, ,x x y y z z< − − − > EQUATION OF A LINE IN A SPACE EQUATION OF A LINE PASSING THROUGH A GIVEN POINT AND PARALLEL TO A GIVEN VECTOR Vector form: Let the line passing through the given point A with position vector a → and let it be parallel to vector b → . i.e. . AP bλ → → = BUT AP OB OA → → → = − b r aλ → → → ⇒ = − ⇒ r a bλ → → → = + , this is vector equation of a line. Cartesian form: Let the given point be A(( 1 1 1, ,x y z ) and <a, b ,c> be the direction ratio & the point P( 1 1 1, ,x y z ), then 1 1 1x x y y z z a b c − − − = = is symmetrical form of line.
  2. 2. EQUATION OF A STRAIGHT LINE PASSING THROUGH TWO GIVEN POINTS Vector form: ( )r a b aλ → → → → = + − Cartesian form: 1 1 1 2 1 2 1 2 1 x x y y z z x x y y z z − − − = = − − − ANGLE BETWEEN TWO LINES: Let 1 2&L L be two lines passing through the origin and with D.R. 1 1, 1,a b c & 2 2 2, ,a b c . Let P be a point on 1L & Q on 2L Therefore the angle θ is given by 1 2 1 2 1 2 2 2 2 2 2 2 21 1 1 2 2 | | a a b b c c Cos a b c a b c θ + + = + + + + Vector form: Let the vectors equation of two lines be 1 1r a bλ → → → = + & 2 2r a bµ → → → = + Cosθ = 1 2 1 2 . | || | b b b b → → → → Condition of perpendicularity: If the lines 1b → and 2b → are perpendicular then 1 2.b b → → =0 Condition of parallelism: If the lines 1b → and 2b → are parallel then 1b → =λ 2b → Cartesian form: Let the Cartesian equation of two lines be 1 1 1 1 1 1 x x y y z z a b c − − − = = & 1 1 1 2 2 2 x x y y z z a b c − − − = = then 1 2 1 2 1 2 2 2 2 2 2 2 21 1 1 2 2 a a b b c c Cos a b c a b c θ + + = + + + + Condition of perpendicularity: 90θ = g i.e. 1 2 1 2 1 2a a bb c c+ + Condition of parallelism: 0θ = i.e 1 1 1 2 2 2 a b c a b c = = SHORTEST DISTANCE Vector form: Let 1 1r a bλ → → → = + & 2 2r a bµ → → → = + be two non interesting lines. Then the shortest distance between the given lines is equal to 1 2 2 1 1 2 ( ).( ) | | | | b b a a b b → → → → → → × − × Cartesian form: Let the lines be 1 1 1 1 1 1 x x y y z z a b c − − − = = and 1 1 1 2 2 2 x x y y z z a b c − − − = = Shortest distance = 2 1 2 1 2 1 1 1 1 2 2 2 2 2 2 1 2 2 1 1 2 2 1 1 2 2 1( ) ( ) ( ) x x y y z z a b c a b c b c b c c a c a a b a b − − − − + − + −
  3. 3. Note: If the lines are intersecting ⇒ lines are Coplanar ⇒ S.D = 0 ⇒ ( 1 2 2 1).(b b a a → → → → × − ) = 0 or 2 1 2 1 2 1 1 1 1 2 2 2 x x y y z z a b c a b c − − − =0 SKEW LINES : Two straight lines in space which are neither parallel nor intersecting are called Skew lines. SHORTEST DISTANCE BETWEEN TWO PARALLEL LINES The shortest distance between two parallel lines 1r a bλ → → → = + & 2r a bµ → → → = + is given by d = 2 1(a -a ) b |b| → → → → × PLANES A Plane is a surface such that if any two distinct points are taken on it then the line containing these points lie completely in it. i.e. every point of the line in it. Or in short A line in the space is called a plane. NOTE: A plane is determined uniquely if any one of the following is known: a) The normal to the plane and its distance from the origin is given. i.e. equation of plane in normal form. b) It passes through a point and is perpendicular to given direction c) It passes through three non collinear points DIFFERENT FORMS OF EQUATION OF PLANES: EQUATION OF PLANE IN NORMAL FORM: Let the Plane ABC be at a distance d from the origin. ON is the normal to the plane in direction n ∧ . Equation of plane is r → .n ∧ =d where d= | |n → p If l, m, n are the direction cosines of the normal to the plane which is at distance d from origin. The equation of plane is lx +my +nz =d NOTE: general form of equation of plane are r → . N → =D & Ax +By +Cz +D=0 EQUATION OF A PLANE PASSING THROUGH A GIVEN POINT & PERPENDICULAR TO A GIVEN DIRECTION Vector form: ( )r a → → − . n → =0 Cartesian form: 1 1 1( ) ( ) ( ) 0A x x B y y C z z− + − + − =
  4. 4. PLANES THROUGH THE INTERSECTION OF TWO PLANES Vector form :Let 1p and 2p be two planes with equations 1 1.r n d → ∧ = and 2 2.r n d → ∧ = . Then equation of plane passing through the intersection of two planes is 1 2 1 2.( )r n n d dλ λ → → → + = + Cartesian form: let 1p and 2p be two planes with equations 1 1 1 1 1 0p a x b y c z d= + + + = & 2 2 2 2 2 0p a x b y c z d= + + + = be two intersecting planes, then 1 2 0p pλ+ = represent a family of planes. EQUATION OF PLANE PASSING THROUGH 3 NON COLLINEAR POINTS Vector form: let a plane passing through three given opoints A,B,C with positions vectors a → , b → , c → . Then equation of plane is ( )r a → → − . ( ) ( ) 0b a c a → → → →   − × − =   Cartesian form: Let the plane pass through the points A ( 1 1 1, ,x y z ), B 2 2 2( , , )x y z ,C 3 3 3( , , )x y z . be any point. Let P( , , )x y z be any point. Then equation of plane is 1 1 1 2 1 2 1 2 1 3 1 3 1 3 1 x x y y z z x x y y z z x x y y z z − − − − − − − − − =0 INTERCEPT FORM OF THE EQUATION OF PLANE: The equation of plane in intercept form is 1 x y z a b c + + = Intersection of two planes: Let 1p and 2p be two intersecting planes with equations 1 1.r n d → ∧ = and 2 2.r n d → ∧ = .and a → be the position vector of any point common to them. r a bλ → → → = + where λ is real number is the vector equation of straight line. NOTE: Whenever two planes intersect, they always intersect along a straight line. ANGLE BETWEEN TWO PLANES: The angle between planes is defined as the angle between their normals. If 1n → and 2n → are normals to the planes and θ be the angle between planes 1 1.r n d → ∧ = and 2 2.r n d → ∧ = . Then 1 2 1 2 . | || | n n Cos n n θ → → → → = NOTE: The planes are perpendicular to each other if 1 2.n n → → =0 and parallel if 1 2n n → → P . Cartesian form: Let θ be the angle between the planes 1 1 1 1 0a x b y c z d+ + + = and 2 2 2 2 0a x b y c z d+ + + = then 1 2 1 2 1 2 2 2 2 2 2 2 1 1 1 2 2 2 a a bb c c Cos a b c a b c θ + + = + + + + NOTE: Two planes are perpendicular if 0 =90θ . i.e. 1 2 1 2 1 2a a bb c c+ + =0
  5. 5. Two planes are parallel if 1 1 1 2 2 2 a b c a b c = = DISTANCE OF A POINT FROM A LINE: Vector form: The length p of the perpendicular drawn from the point p with position vector a → to the plane r → . n → =d is given by p= | . | | | a n d n → → → − NOTE: the length of perpendicular from origin to plane r → . n → =d is given by p= | | | | d n → Cartesian form: The length p of the perpendicular drawn from the point P( , , )x y z to the plane Ax+By+Cz+D=0 is given by p= 1 1 1 2 2 2 Ax By Cz D A B C + + + + + ANGLE BETWEEN A LINE AND A PLANE: If the equation of line is r a bλ → → → = + and equation of plane is r → . n → =d . Then the angle θ between line and normal to plane is . | || | b n Cos b n θ → → → → = So angle φ between line and plane is 90-θ .i.e. (90 )Sin Cosθ θ− = i.e. Sinφ = . | || | b n b n → → → → NOTE: If we have Cartesian form change it into vector form.
  6. 6. Two planes are parallel if 1 1 1 2 2 2 a b c a b c = = DISTANCE OF A POINT FROM A LINE: Vector form: The length p of the perpendicular drawn from the point p with position vector a → to the plane r → . n → =d is given by p= | . | | | a n d n → → → − NOTE: the length of perpendicular from origin to plane r → . n → =d is given by p= | | | | d n → Cartesian form: The length p of the perpendicular drawn from the point P( , , )x y z to the plane Ax+By+Cz+D=0 is given by p= 1 1 1 2 2 2 Ax By Cz D A B C + + + + + ANGLE BETWEEN A LINE AND A PLANE: If the equation of line is r a bλ → → → = + and equation of plane is r → . n → =d . Then the angle θ between line and normal to plane is . | || | b n Cos b n θ → → → → = So angle φ between line and plane is 90-θ .i.e. (90 )Sin Cosθ θ− = i.e. Sinφ = . | || | b n b n → → → → NOTE: If we have Cartesian form change it into vector form.

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