Three Dimensional Geometry

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1 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 ) and (x, y, z ) are x x, y y, z z Let a, b, c be the direction ratio of a line whose direction cosines are l, m, n the l = m = n l + m + n = l = ± a a + b + c m = ± b a + b + c n = ± c a + b + c If line makes angles α, β, γ with coordinate axes then l = cosα m = cosβ n = cosγ cos α + cos β + cos γ = Vector equation of a straight line passing through a fixed point with the position vector a and to given vector b r = a +λb where λ is parameter and r = xiˆ + yj ˆ + zkˆ Cartesian equation of straight line passing through a fixed point (x, y, z ) having direction ratio x x y y z z (a, b, c) is given by x x y y z z The coordinates of any point on the line = λ are (x + aλ, y + bλ, z + cλ) where λ R Angle between two lines whose direction rahos are a, b, c and (a, b, c ) is given by aa + bb + cc cosθ = a + b + c a + b + c If lines are perpendicular then a a + b b + c c = 0 a b c If lines are parallel then Angle between tow lines : r = a + λ m and r = a + µ m is given as cosθ = lines are perpendicular if m. m = 0. Lines are parallel m =λm m. m m m and so, two

2 Skew lines : Lines in space, which are neither parallel, nor intersecting are called skew lines, such pair of lines are non-coplanar. Shortest distance : r = a + λ b and r = a + µ b are two skew lines, then distance d between them is given by d = ( a a ) ( b b ) b b Distance d between parallel line : if r = a + λ b and r = a + µ b are two parallel lines, then distance d, between them is given by d = ( a a ) b b Plane : A plane is uniquely determined if any one of the following is known: (i) The normal to the plane and its distnace from origin. (ii) It passes through a given point and is perpendicular to a given direction. (iii) It passes through three given non collinear points. Equation of a plane at a distance p from origin and normal to vector n is given by r. n ˆ = p or lx + my + nz = p. Where l, m, n be the direction cosines of normal to the plane vector. General equation of plane passing through a point a and having normal vector to plane as n is ( r a). n = 0. Corresponding Cartesian form is a(x x ) + b(y y ) + c(z z ) = 0, where a, b, c are direction ratios of normal to plane. General equation of plane which cuts off intercepts a, b and c on x, y and z-axis respectively is x + y + z =. Equation of plane passing through three non-collinear points a, b, c is ( r a) ( b a) ( c a) = 0. Corresponding Cartesian from is x x y y z z x x y y z z = 0 x x y y z z where (x, y, z ), (x, y, z ) and (x 3, y 3, z 3 ) are co-ordinates of known point. Angle between two planes is defined as the angle between their normal cosθ = aa + bb + cc a + b + c a + b + c or cosθ = where n & n are vectors normal to planes or a, b, c and a, b, c are dr s of normal to planes. Note : if aa + bb + cc = 0 planes are to each other a b c planes are parallel to each other n. n n n

3 Let r = a + λb and r = a + λ b be two lines then these lines are coplanar of ( a a ).( b b ) = 0 Length of perpendicular from a point having a position vector a to the plane rn. = d is given by an. n d Length of perpendicular from a point P(x y z ) to the plane ax + by + cz + d = 0 is given by ax + by + cz + d a + b + c Angle between line r = a + λ b and plane rn. = d is the complementary to the angle between line bn. and normal to the plane is given by sin b n line will be parallel to plane if bn=. 0 and line will be perpendicular to plane if b =λ n. P = ax + b y + cz + d = 0 and P = ax + b y + cz + d = 0 be two intersecting (non parallel planes) then ax b y cz d k ax b y cz d P kp ( ) + ( ) = 0( + = 0) for different real values of k, represents a family of planes passing through the line of intersection of the planes P = 0 P = 0. Note : The equation P + kp = 0 represents all members of above sold family except the plane P = 0. However if plane P is needed take the eq. of family P + kp = 0 Question for Practice Very Short Answer Type Questions ( Mark). Write intercept cut off by the plane x + y z = on x axis Write the vector equation of a line + z Write the direction cosine of line joining (, 0, 0) and (0,, ) 4. What are the direction cosine of a line which makes equal angles with coordinate axes. 5. What is the cosine of the angle which vector iˆ + ˆj + kˆ makes with y axis 6. What is the distance of the following plane from origin x y + z + = Write the distance of the point (a, b, c) from x axis. x y z 3 8. For what value of λ the line its perpendicular to tbe plane 3x y z = 7. 9 λ 6 9. If a line makes α, β, γ and with the x axis, y axis and z axis respectively. Find the value of sin α + sin β + sin γ. 0. Find the coordinate of the pt where line x 3 y 5 z crosses the yz plane. 3 5

4 . x y 5 z + Write the direction ratio of the line. 3. What is the equation of plane parallel to XOY plane and passing through (3, 4, 8) 3. Write the direction cosines of the perpendicular from the origin to the plane r ( iˆ + ˆj + kˆ ) = Write the direction ratio of the normal of plane x + y + z = Write the value of λ for which the plane x 4y + 3z = 7 and x + y + λz = 8 are to each other. 6. Write the vector eq. of plane x + y + z = x y z Write the equation of line to the line and passes through (0, 0, ) What is the equation of line passes through (,, ) and to the plane x + y + z = If a line makes an angle 60, 30, 90 with the positive direction of x, y, z axis respectively then write the direction cosines of line. 0. What is the equation of a plane that cut the coordinate axes at (a, 0, 0), (0, b, 0) and (0, 0, c).. Write the angle between line Short Answer Type Questions (4 Marks) x y + z 3 and the plane 3x + 4y + z + 5 = Find the equation of the line passing through the points (,, ) and (3,, ). At what point it meet yz plane.. Find the equation of the line passing through the point (, 3, ) and perpendicular to the line x y z and 3 x + y z x y z 3 x y 4 z 5 3. Find the shortest distance between the lines and Show that the four points (0,, 0) (,, ) (,, ) and (3, 3, 0) are coplanar & also find the equation of plane containing these point. 5. A variable plane which remains a constant distance 3P from the origin cuts the coordinate axis A, B and C show that the locus of the centroid of ABC is x + y + z = P 6. Find the equation of the plane passing through pt (, 3, 4) and to the plane 5x 6y + 7z =3. 7. Find the equation of the plane passing through the points (,, ) and (9, 3, 6) and perpendicular to the plane x + 6y + 6z =. 8. Find the equation of plane passing through origin and perpendicular to each of the plane x + y z = and 3x 4y + z = Find the distance between two parallel planes x y + 3z + 4 = 0 6x 3y + 8z 3 = 0 0. Find the equation of plane which contains the line of intersection of the plane r ( iˆ + ˆj + 3 kˆ ) = 4 and r ( iˆ + ˆj + kˆ ) = 5 and which it to the plane r (5iˆ + 3 ˆj 6 kˆ ) = 8.

5 . Find the distance of the point A(, 5, 0) from the point of intersection of the line r = ( iˆ ˆj + kˆ ) + λ (3iˆ + 4 ˆj + kˆ ) and the plane r ( iˆ ˆj + kˆ) = 5. Find the distance of the point (,, 3) from the plane x y + z = 5 measured along a line x y z parallel to Find the distance of the point A(, 3, 4) from the line to the plane 4x + y 3z + = Find a points on the line x + y + 3 3z + 4 measured parallel x + y + z 3 at a distance of 5 units from the point (, 3, 3) Find the equations of the two lines through the origin which intersect the line at angle of 3 π. x + y 3 z + 6. Show that the lines and 3 of plane containing them. x 3 y 3 z x y 7 z + 7 are coplanar. Also find the equation 3 7. Show that the line r = ( iˆ ˆj + 3 kˆ ) + λ ( iˆ ˆj + 4 kˆ) is to the plane 5 j + k) = 5. Also find the distance between them. 8. Find the value of λ so that the lines to each other. x 7 y 4 5 z 0 3 λ and 7 7 x y 5 6 z 3λ 5 are 9. Show that the plane whose vector equation is r ( iˆ + ˆj kˆ ) = 3 contains the line whose vector equation is r = ( iˆ + ˆj ) + λ (iˆ + ˆj + 4 kˆ ). 0. If the point (,, P) and ( 3, 0, ) be equidistant from plane r (3iˆ + 4 ˆj kˆ ) + 3 = 0 find value of P.. Considering the earth as a plane having equation 5x + 9y 0z + 38 = 0. A monument is standing vertically such that its peak is at the point (,, 3). Find the height of monument. How can we save our monument. Long Answer Type Questions (6 Marks). A line makes α, β, γ, δ with the four diagonals of a cube prove that cos α + cos β + cos γ + cos δ = Show that angles between any two diagonals of cube is cos 3.

6 3. If l, m, n and l, m, n be the direction cosines of two mutually prependicular lines. Show that direction cosines of the line to both of them are (m n m n ), (n p n p ), (l m l m ). 4. Find the foot of the perpendicular from the point (0,, 3) on the line Also find the length of perpendicular. 5. Find the image of the point (, 6, 3) on the line x y z 3 x + 3 y z x 4 y + 3 z + x y + z Prove that the lines and intersect. Also find the cordinates of their point of intersection. 7. Find the shortest distance between the lines r = ( + λ ) iˆ + ( λ ) ˆj + ( + λ) kˆ r = ( + µ ) iˆ ( µ ) ˆj + ( + µ ) kˆ 8. Find the length and the foot of the prependicular from the point (7, 4, 5) to the plane x + 4y z =. 9. Find the image of the point (, 3, 4) on the plane x y + z + 3 = Find the equation of a plane passing through the points (0, 0, 0) and (3,, ) and to the x 4 y + 3 z + line Find the equation of the plane passing through the point (0, 7, 7) and containing the line x + y 3 z Two bikes are running at the speed more than allowed speed on the road along the lines (3 i j) i + 3 k) Using shortest distance formula check whether they meet to an accident or not? While driving, should driver maintain the speed limit. Justify. 3. Find the equation of the plane determined by the point A(3, ) B(5,, 4) and C(,, 6). Also find distance of the point P(6, 5, 9) from plane. 4. Find the coordinates of the point where the line through (3, 4, 5) and (, 3, ) crosses the plane determined by the points A(,, 3) B(,, ) C(, 3, 6). ANSWERS Very Short Answer ( Mark). 6. r = (5 iˆ 4 ˆj + 6 kˆ ) + λ (3iˆ + 7 ˆj + kˆ )

7 3. 5.,, cosθ= b 4. ±, ±, ± c , 3,. z = 8 9 0, 3.,, 4. (,, ) λ = 6. r.( iˆ + ˆj + kˆ ) = x y z ,,0 x y z x y z =. sin 7 9 Short Answer (4 Mark) ,, 3. + y z x 3y + z = x 6y + 7z = x + 4y 5z = 9 8. x + y + 5z = r.(33iˆ + 45 ˆj + 50 kˆ ) = (,, 3) and (4, 3, 7) 5. x y z x y z and 7 sq units x 3 y 3 z Hint : Give line = λ Any pt on line (λ + 3, λ + 3, λ) DR of op (λ + 3 0, λ + 3 0, λ 0) line op makes 3 π with line pq

8 π (3+ λ ) + (3 + λ ) + λ cos = (3+ λ ) + (3 + λ ) + λ λ =, or eq of requred line x y z x y z, 6. x + y + z = λ λ + 8λ + 8 x 0 y 0 z 0 and unit 8. λ = 7 0. P =, 7 3 Answer (6 Mark) 4. Foot of (, 3 ) length of 5. ( p, 7) 6. Given line intersect at (5, 7, 6) Foot of lenght of 3 0. x 9y z = 0. x + y + z = 0. S.D is zero it means they meet an accident 3. 3x 4y + 3z 9 = 0 4. (,, 7) 6 34 unit. x 0 y 0 z

THREE DIMENSIONAL GEOMETRY

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 1 1 1 (i) The coordinates of point