VIEWS: 17 PAGES: 3 CATEGORY: Algebra POSTED ON: 9/14/2012 Public Domain
1 Three –Dimensional Geometry (1 mark questions) 1. Find the vector equation of the line through the point (5,2,-4) and which is parallel to the vector 3î + 2ĵ - 8k. 2. Find the distance of the plane 2x-3y+4z-6=0 from the origin. 3. Find the angle between two planes: 3x - 6y + 2z = 7 and 2x + 2y – 2z = 5 4. A straight line makes angles 60˚ and 45˚ with the positive direction of x – axis and y – axis respectively. What angle does it make with the z – axis ? 5. Find the equation of the plane passing through the point (-1,0,7) and parallel to the plane 3x - 5y + 4z = 11. 6. Find the distance from the point (2,5,-3) to plane 6x – 3y +2z = 4. 7. Show that the normals to the planes r . (î – ĵ + k) = 4 and r .( 3î + 2ĵ - k) -11 = 0 are perpendicular to each other. 8. Write the equation of the plane whose intercepts on the co- ordinate axis are -4, 2 and 3. 9. If the line r = (î - 2ĵ + k)+λ(2î + ĵ + 2k) is parallel to the plane r .( 3î - 2ĵ + mk) = 14, find the value of m. 10. If 4x + 4y – λz =0 is the equation of the plane through the origin that contains the line x–1 = y+1 = z , 2 3 4 find the value of ‘λ’. SECTION - B(4 marks) 1. Find the shortest distance between the lines l1 and l2 whose vector equations are :- r = î + ĵ + λ(2î – ĵ + k) r = 2î + ĵ – k + μ(3î - 5ĵ + 2k) 2. Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z =0. 3. Find the vector equation of the line passing through (1,2,3) and parallel to the planes r. (î – ĵ + 2k) = 5 and r .( 3î + ĵ + k ) = 6. 4. Find the vector equation of the plane passing through the points A(1,-2,5) , B(0,-5,-1) and C(-3,5,0). 5. Find the image of the point (3,-2,1) in the plane 3x - y + 4z = 2. 2 6. Find the image of the point (1,6,3) in the line x = y – 1 = z – 2 . 1 2 3 7. If l1, m1, n1 ; l2 , m2 , n2 are the direction cosines of two mutually perpendicular lines. Show that the direction cosines of the line perpendicular to both of them are m1n2 – m2n1 , n1l2 – n2l1 , l1m2 – l2m1. 8. Verify that l1+l2+l3 , m1+m2+m3 , n1+n2+n3 can be taken as the direction cosines of a line equally √3 √3 √3 inclined to three mutually perpendicular lines with direction cosines l1,m1,n1;l2,m2,n2 and l3,m3,n3. SECTION – C(6 Marks) 1. A line makes angles α, β, γ and δ with the diagonals of a cube, prove that cos2α + cos2β + cos2γ + cos2δ = 4 3 2. Show that the lines r1 = î + ĵ – k + λ(3î - ĵ) and r2 = 4î – k + μ(2î + 3k) intersect. Find their point of intersection. 3. Find the equation of the plane through the intersection of the planes r .(î + 3ĵ)+6 = 0 and r .(3î - ĵ - 4k) = 0 which is at a unit distance from the origin. 4. Show that the lines : x - 1 = y – 3 = z and x – 4 = y – 1 = z – 1 2 4 -1 3 -2 1 are coplanar. Also find the equation of the plane containing these lines. 5. If a variable line in two adjacent positions has direction cosines l, m, n and l+δl , m+δm , n+δn , show that the small angle δθ between two positions is given by:- (δθ)2 = (δl)2 + ( δm)2 + ( δn)2 6. Find the equations of the line of shortest distance between the lines : x – 8 = y +9 = z-10 and x-15 = y-29 = 5-z 3 -16 7 3 8 5 also find the shortest distance between the lines. 7. A variable plane which remains at a constant distance 3p from the origin cuts the co-ordinate axes at A , B , C. Show that the locus of the centroid of ΔABC is x -2 + y -2 +z -2 = p-2 . 8. Find the equation of the perpendicular from the point (3,-1,11) to the line x = y-2 = z-3 . 2 3 4 Also find the foot of perpendicular and the length of perpendicular. 9. Find the distance of the point (1,-2,3) from the plane x - y + z = 5 measured parallel to the line x+1 = y+3 = z+1 . 2 3 -6 3 Answers Three Dimensional Geometry 1 MARKS 1) r = (5 î + 2ĵ-4 k ) + λ(3 î + 2ĵ- 8k ) 2) 6 √29 3) cos-1 5√3 4) 60º 5) 3x - 5y + 4z – 25 = 0 21 6) 13 units 8) -3x + 6y + 4z = 12 9) m = -2 7 10) λ= 5 4 MARKS 1) 10 2) x – z + 2 = 0 √59 3) r = î + 2ĵ + 3k + λ(-3î + 5ĵ + 4 k ) 4) r .( 3î + ĵ – k ) = - 4 5) (0,-1,-3) 6) (1,0,7) 6 MARKS 2) (4,0,-1) 3) r .( -2î + 4ĵ + 4k ) + 6 = 0 or 4) 2x – 5y -16z +13 = 0 r . (4î + 2ĵ -4 k ) + 6 = 0 6) x-5 = y – 7 = z-3 , 14 units 8)(2,5,7) , √53 units 2 3 6 x-3 = y + 1 = z-11 -1 6 -4 9) 1 unit