THREE DIMENSIONAL GEOMETRY

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1 For more important questions visit : www4onocom CHAPTER 11 THREE DIMENSIONAL GEOMETRY POINTS TO REMEMBER Distance between points P(x 1 ) and Q(x, y, z ) is PQ x x y y z z (i) The coordinates of point R which divides line segment PQ where P(x 1 ) and Q(x, y, z ) in the ratio m : n internally are (ii) mx nx my ny mz nz,, m n m n m n The co-ordinates of a point which divides join of (x 1 ) and (x, y, z ) in the ratio of m : n externally are mx nx my ny mz nz,, m n m n m n Direction ratios of a line through (x 1 ) and (x, y, z ) are x x 1, y y 1, z z 1 Direction cosines of a line whose direction ratios are a, b, c are given by l, m, n a b c a b c a b c (i) Vector equation of a line through point a and parallel to vector b is r a b (ii) Cartesian equation of a line through point (x 1 ) and having direction ratios proportional to a, b, c is XII Maths 111

2 x x y y z z (i) Vector equation of line through two points a and b is r a b a (ii) Cartesian equation of a line through two points (x 1 ) and (x, y, z ) is x x y y z z x x y y z z Angle between lines r a 1 b 1 and r a µ b is given b 1 b by cos b b 1 x x Angle between lines 1 y y 1 z z 1 x x and a y y z z b c is given by a a b b c c cos Two lines are perpendicular to each other if b 1 b 0 Equation of plane : or a 1 a + b 1 b + c 1 c = 0 (i) (ii) (iii) At a distance of p unit from origin and perpendicular to n is r n p and corresponding Cartesian form is lx + my + nz = p when l, m and n are dcs of normal to plane Passing through a and normal to n is r a n 0 and corresponding Cartesian form is a(x x 1 ) + b (y y 1 ) + c(z z 1 ) = 0 where a, b, c are dr s of normal to plane and (x 1 ) lies on the plane Passing through three non collinear points is r a b a c a 0 11 XII Maths

3 x x y y z z or x x y y z z x x y y z z (iv) H a vin g in te rce p ts a, b and c on co-ordinate axis is x y z 1 (v) Planes passing through the line of intersection of planes r n 1 d 1 and r n d is r n 1 d 1 r n d 0 (i) Angle between planes r n 1 d 1 and r n d is n 1 n given by cos n n 1 (ii) Angle between a 1 x + b 1 y + c 1 z = d 1 and a x + b y + c z = d is given by a 1a b 1b c 1c cos a 1 b 1 c 1 a b c (iii) Two planes are perpendicular to each other iff n 1 n 0 or a 1 a + b 1 b + c 1 c = 0 (iv) Two planes are parallel iff n 1 n for some scaler 0 or a b c a b c (i) Distance of a point a from plane r n d is (ii) a n d n XII Maths 113 Distance of a point (x 1 ) from plane ax + by + cz = d is ax 1 by 1 cz 1 d a b c 1 (i) Two lines r a 1 b 1 and r a b are coplanar Iff a a 1 b 1 b 0 and equation of plane, r a b b containing these lines is 1 1 0

4 (ii) x Two lines x y y z z and a 1 b 1 c 1 x x y y z z are coplanar Iff x x y y z z and equation of plane containing them is x x y y z z 0 (i) The angle between line r a b and plane r n d is given as sin b n b n 0 b 90 (ii) The angle between line a x + b y + c z = d is given as x x 1 y y 1 z z 1 and plane a a b b c c sin (iii) A line r a b is parallel to plane r n d b n 0 or a 1 a + b 1 b + c 1 c = 0 r = a + b r n = d 114 XII Maths

5 VERY SHORT ANSWER TYPE QUESTIONS (1 MARK) 1 What is the distance of point (a, b, c) from x-axis? What is the angle between the lines x = 3y = z and 6x = y = 4z? 3 Write the equation of a line passing through (, 3, 5) and parallel to line x 1 y z Write the equation of a line through (1,, 3) and perpendicular to r i j 3 k 5 x What is the value of for which the lines y z and 5 x 1 y z are perpendicular to each other 3 6 If a line makes angle,, and with co-ordinate axes, then what is the value of sin + sin + sin? r i j j k into Cartesian form 7 Write line 8 If the direction ratios of a line are 1,, then what are the direction cosines of the line? 9 Find the angle between the planes x 3y + 6z = 9 and xy plane 10 Write equation of a line passing through (0, 1, ) and equally inclined to co-ordinate axes 11 What is the perpendicular distance of plane x y + 3z = 10 from origin? 1 What is the y-intercept of the plane x 5y + 7z = 10? 13 What is the distance between the planes x + y z + = 0 and 4x + 4y z + 5 = 0 14 What is the equation of the plane which cuts off equal intercepts of unit length on the coordinate axes 15 Are the planes x + y z + 4 = 0 and 3x + 3y 6z + 5 = 0 intersecting? 16 What is the equation of the plane through the point (1, 4, ) and parallel to the plane x + y 3z = 7? XII Maths 115

6 17 Write the vector equation of the plane which is at a distance of 8 units from the origin and is normal to the vector i j k 18 What is equation of the plane if the foot of perpendicular from origin to this plane is (, 3, 4)? 19 Find the angles between the planes 1 and r 3i 6j k 0 r i j k 0 What is the angle between the line plane x + y z + 4 = 0? x 1 y 1 z and the If O is origin OP = 3 with direction ratios proportional to 1,, then what are the coordinates of P? What is the distance between the line r i j 3k i j 4k from the plane r i 5 j k Write the line x = 3y = 4z in vector form SHORT ANSWER TYPE QUESTIONS (4 MARKS) 4 The line x 4 y 4 k z 1 x 4y + z = 7 Find the value of k lies exactly in the plane 5 Find the equation of a plane containing the points (0, 1, 1), ( 4, 4, 4) and (4, 5, 1) Also show that (3, 9, 4) lies on that plane 6 Find the equation of the plane which is perpendicular to the plane r 5i 3 j 6k 8 0 & which is containing the line of intersection of the planes r i j 3k and r i j k If l 1, m 1, n 1, and l, m, n are direction cosines of two mutually perpendicular lines, show that the direction cosines of line perpendicular to both of them are m 1 n n 1 m, n 1 l l 1 n, l 1 m m 1 l 116 XII Maths

7 8 Find vector and Cartesian equation of a line passing through a point with position vectors i j k and which is parallel to the line joining the points with position vectors i 4 j k and i j k 9 Find the equation of the plane passing through the point (3, 4, ) and (7, 0, 6) and is perpendicular to the plane x 5y = Find equation of plane through line of intersection of planes r i 6 j 1 0 and r 3i j 4k 0 which is at a unit distance from origin 31 Find the image of the point (3,, 1) in the plane 3x y + 4z = 3 Find the equation of a line passing through (, 0, 5) and which is parallel to line 6x = 3y + 1 = z 33 Find image (reflection) of the point (7, 4, 3) in the line x y 1 z Find equations of a plane passing through the points (, 1, 0) and (3, 4, 5) and parallel to the line x = 3y = 4z 35 Find distance of the point ( 1, 5, 10) from the point of intersection of line x y 1 z and the plane x y + z = 5 36 Find equation of the plane passing through the points (, 3, 4) and (1, 1, 3) and parallel to the x axis 37 Find the distance of the point (1,, 3) from the plane x y + z = 5, x y z measured parallel to the line Find the equation of the plane passing through the intersection of two plane 3x 4y + 5z = 10, x + y 3z = 4 and parallel to the line x = y = 3z 39 Find the distance between the planes x + 3y 4z + 5 = 0 and r 4 i 6 j 8 k Find the equations of the planes parallel to the plane x y + z 3 = 0 whose perpendicular distance from the point (1,, 3) is 1 unit XII Maths 117

8 41 Show that the lines x y 1 3 intersection 4 x 1 y 3 z 5 and z 5 4 Find the shortest distance between the lines r l j 3k i 3 j 4k and r i 4j 5k 3i 4j 5k 6 intersect each other Find the point of 43 Find the distance of the point (, 3, 4) from the line y 3 3z 4 measured parallel to the plane 4x + 1y 3z + 1 = Find the equation of plane passing through the point ( 1, 1, ) and perpendicular to each of the plane r i 3j 3k and r 5i 4j k 6 45 Find the equation of a plane passing through ( 1, 3, ) and parallel to x y z each of the line and 1 3 x y 1 z Show that the plane r i 3 j 5k 7 contains the line r i 3j 3k 3i j x 3 LONG ANSWER TYPE QUESTIONS (6 MARKS) 47 Check the coplanarity of lines r 3i j 5 k 3i j 5k r i j 5 k µ i j 5k If they are coplanar, find equation of the plane containing the lines 48 Find shortest distance between the lines : x 8 y 19 z 10 x 15 y 9 z 5 and XII Maths

9 49 Find shortest distance between the lines : r 1 i j 3 k r i j k A variable plane is at a constant distance 3p from the origin and meets the coordinate axes in A, B and C If the centroid of ABC is (), then show that + + = p 51 A vector n of magnitude 8 units is inclined to x axis at 45, y axis at 60 and an acute angle with z-axis If a plane passes through a point, 1,1 and is normal to n, find its equation in vector form 5 Find the foot of perpendicular from the point i j 5k on the line r 11i j 8k 10i 4 j 11k Also find the length of the perpendicular 53 A line makes angles,,, with the four diagonals of a cube Prove that 4 cos cos cos cos 3 54 Find the equation of the plane passing through the intersection of planes x + 3y z = 1 and x + y z + 3 = 0 and perpendicular to the plane 3x y z = 4 Also find the inclination of this plane with xy-plane ANSWERS 1 b c x y 3 z r i j 3k i j 3k 5 = 6 7 x 1 y 1 z ,, XII Maths 119

10 9 cos 1 (6/7) 10 x y 1 z, a a a a R x + y + z = 1 15 No 16 x + y 3z = 8 17 r i j k 4 18 x + 3y + 4z = cos (line is parallel to plane) 1 ( 1,, ) 3 r o 6 i 4j 3k k = 7 5 5x 7y + 11z + 4 = 0 6 r 51 i 15j 50k r i j k i j k and 9 x y + 3z = 1 x y 1 z r 8i 4j 8k 1 0 or r 4i 8j 8k (0, 1, 3) 3 x y z ,, x 7y z = y + 4z = 5 10 XII Maths

11 x 0y + 7z = units 9 40 x y + z = 0 or x y + z = ,, r 9i 17j 3k 0 45 x 7y + 4z + 15 = 0 47 x y + z = r i j k 5 i j 3 k, x 13y 4z 9, cos For more important questions visit : www4onocom XII Maths 11

Three Dimensional Geometry

Three Dimensional Geometry Teaching learning points Distance between two given points P(x, y, z ) and Q(x, y, z ) is PQ = ( x x ) + ( y y ) + ( z z ) Direction ratio of line joining the points (x, y, z