THREE DIMENSIONAL GEOMETRY KEY POINTS TO REMEMBER
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1 THREE DIMENSIONAL GEOMETRY KEY POINTS TO REMEMBER Line in space Vector equation of a line through a given point with position vector vector is. Vector equation of line through two points with position vectors, = ) is and parallel to the Angle θ between lines and is given by cosθ = Two lines are perpendicular to each other if = 0. Plane Equation of a plane at a distance of d unit from origin and perpendicular to is = d. Equation of plane passing through and normal to is. = 0. Equation of plane passing through three non collinear points with position vectors is ( = 0. If three points are collinear, there are infinitely many possible planes passing through them. Planes passing through through the intersection of planes = and =d 2 is given by ( ) + λ ( ) = 0. Angle between the two planes = and =d 2 is given by cos θ =. Two planes are perpendicular to each other iff = 0. Two planes are parallel iff = λ for some scalar λ 0. A line is parallel to the plane = d iff ASSIGNMENT 1. Show that the four points (0, -1, -1), (-4, 4, 4), (4, 5, 1) and (3, 9, 4) are coplanar. Find the equation of plane containing them. 2. Find the equation of plane passing through the points (2, 3, 4), (-3, 5, 1) and (4, -1, 2). 3. Find the equation of the plane through the points (2, 1, -1) and (-1, 3, 4) and perpendicular to the plane x - 2y + 4z = Show that the following planes are at right angles:. (2 ) = 5 and. ( ) = 3 5. Determine the value of p for which the following planes are perpendicular to each other: i.. ( ) = 7 and. ( ) = 26
2 ii. 2x - 4y + 3z = 5 and x + 2y + pz = Find the equation of the plane passing through the point (-1, -1, 2) and perpendicular to the planes 3x + 2y - 3z = 1 and 5x - 4y + z = Obtain the equation of the plane passing through the point (1, -3, -2) and perpendicular to the planes x + 2y + 2z = 5 and 3x + 3y + 2z = Find the equation of the plane passing through the origin and perpendicular to each of the planes x + 2y - z = 1 and 3x - 4y + z = Find the equation of the plane passing through the points (1. -1, 2) and (2, -2, 2) and which is perpendicular to the plane 6x - 2y + 2z = Find the equation of the plane through the point (1, 4, -2) and parallel to the plane -2x + y -3z = Find the vector equations of the planes through the intersection of the planes. (2 )+ 12 = 0 and. ( ) = 0 which are at unit distance from the origin. 12.If the line =( ) + λ ( ) is parallel to the plane. (3 ) =14, find the value of m. 13. Find the equation of the plane through the points (1, 0, -1),(3, 2, 2)and parallel to the line = 14. Find the equation of the plane passing through the point (0, 7, -7) & containing the line = 15. Prove that the lines = and = are coplanar. Also, find the plane containing these two lines. 16. Show that the lines given below are coplanar. Also find the equation of the plane containing them. =( ) + λ ( ) and =(2 ) + ( ) 17. Find the image of the point (3, -2, 1) in the plane 3x - y + 4z = Find the length and the foot of perpendicular from the point (7, 14, 5) to the plane 2x + 4y - z = Find the reflection of the point (1, 2-1) in the plane 3x - 5y + 4z = Find the coordinate of the foot of perpendicular drawn from the point (5, 4, 2) to the line =. Hence or otherwise deduce the length of the perpendicular. 21. Find the image of the point with position vector 3 in the plane. (2 ) = Find the length and the foot of the perpendicular from the point (1, 1, 2) to the plane. ( + 4 ) + 5 = 0.
3 23. The cartesian equation of a line AB is =. Find the direction cosines of a line parallel to AB. 24. Show that the lines = and = intersect. Also find the point of intersection. 25. Find the coordinates of the point where the line through the points (3, -4, -5) and (2, -3, 1) crosses the plane 2x + y + z = Find the vector and cartesian equation s of the planes containing the two lines =( 2 ) + λ ( ) and =( 3 ) + ( ). 27. Find the equation of the plane passing through the point (1, 1, 1) and containing the line =( -3 ) + λ (3 ). Also show that the plane contains the line =( - ) + λ ( ). 28. Two airships are moving in space along the lines = and = An astronaut wants to move from one ship to another ship when the two airships are closest. What is the least distance between the airships? 29. A bird is located at A (3, 2, 8). She wants to move to the plane 3x + 2y + 6z + 16 = 0 in shortest time. Find the distance she covered. 30. By computing the shortest distance determine whether the following pairs of lines intersect or not: = and =. 31. From a point A (2, 3, 8) in space, a shooter aims to hit the target at P(6, 5, 11). If the line of fire is =, what you think about the success of the shooter? 32. An astronaut at A(7, 14, 5) in space wants to reach a point P on the plane 2x + 4y - z = 2 when AP is least. Find the position of P and also the distance AP travelled by the astronaut. 33. Find the distance of the point (3, 4, 5) from the plane x + y + z = 2 measured parallel to the line 2x = y = z. Answer key 1. 5x - 7y + 11z +4 = 0 2. x + y -z = x + 17y + 4z = i. 17 ii x + 9y + 11z - 8 =0 7. 2x - 4y + 3z -8 = 0 8. x + 2y + 5z = 0 9. x + y - 2z + 4 = x - y + 3z + 8 = (2 + 2 )+ 3 = 0 and. ( ) + 3 = m = x - y - 2z - 6 = x + y + z = x - 2y + z = ( ) + 7 = (0, -1, -3) 18. (1, 2, 8); 3 units 19. (73/25, -6/5, 39/25) 20. (1, 6, 0) ; 2 units 21. (1, 2, 1)
4 22. ; (-1/12, 25/12, -2/12) 23.,, 24. (-1, -1, -1) 25. (1, -2, 7) 26.. ( ) - 37 = 0; 10x + 5y -4z - 37 = x - 2y + z = units units 30. lines do not intersect 31. he will be successful 32. P(1, 2, 8); 3 units units VECTOR ALGEBRA IMPORTANT POINTS TO REMEMBER A quantity that has magnitude as well as direction is called a vector. It is denoted by a directed line segment. Two or more vectors which are parallel to same line are called collinear vectors. Position vector of a point P(a, b, c) w.r.t. origin (0, 0, 0) is denoted by, where = a and l l =. If A(x 1, y 1, z 1 ) and B (x 2, y 2, z 2 ) be any two points in space, then l l = = (x 2 - x 1 ) + (y 2 - y 1 ) + (z 2 - z 1 ) and If two vectors and are represented in magnitude and direction by the two sides of a triangle taken in order, then their sum is represented in magnitude and direction by third side of triangle taken in opposite order. This is called triangle law of addition of vectors. If is any vector and is a scaler, then λ is a vector collinear with and lλ l = lλl l l. If and are two collinear vectors, then = λ, where λ is some scaler. Any vector can be written as = l l, where is a unit vector in the direction of. If and be the position vectors of points A and B, and C is any point which divides in the ratio m : n internally then position vector of point C is given as =. If C divides in ratio m : n externally, then =. The angles made by = a +b + c with positive direction of x, y and z-axis are called direction angles and cosines of these angles are called direction cosines of denoted as l = cos, m = cos, n = cos. Also l =, m =, n = & l 2 + m 2 + n 2 = 1 The numbers a, b, c proportional to l, m, n are called direction ratios.
5 Scaler product of two vectors and is denoted as. & defined as. = l ll lcosθ where is the angle between and. (0 ). if and only if is perpendicular to = l l 2, so = 1. If = a 1 + a 2 +a 3, = b 1 + b 2 +b 3, then a 1 b 1 + a 2 b 2 + a 3 b 3. Cross product ( Vector product ) of two vectors and is denoted as x & defined as x = l ll l sinθ,where is the angle between and. (0 ) and is a unit vector perpendicular to both and such that, and form a right handed system. x = iff is parallel to. Unit vector perpendicular to both & = l x l is the area of parallelogram whose adjacent sides are and is the area of parallelogram whose diagonals are and ASSIGNMENT 1. If are the position vectors of the points (1, -1), (-2, m), find the value of m for which are collinear. 2. If a vector makes angles with OX, OY and OZ respectively, prove that + sin 2 = Find the direction cosines of a vector which is equally inclined with OX, OY and OZ. If l l is given, find the total number of such vectors. 4. A vector is inclined at equal angles to OX, OY and OZ. If the magnitude of is 6 units, find 5. A vector has length 21 and d. r.s 2, -3, 6. Find the direction cosines and components of, given that it makes an acute angle with x axis. 6. Find the angles at which the vector is inclined to each of the coordinate axes. 7. For any vector, prove that = (. ) + (. ) + (. ). 8. Find the value of p for which the vectors and + p +3 are i. Perpendicular ii. Parallel 9. If are unit vectors inclined at an angle, then prove that sin = l l. 10. If makes equal angles with & and has magnitude 3, then prove that the angle between and each of & is cos -1 (1/ ).
6 11. If + + =, l l = 3, l l = 5 and l l = 7, find the angle between. 12. Find a vector of magnitude 9, which is perpendicular to both the vectors & Find a unit vector perpendicular to the plane ABC where A, B, C are the points (3, -1, 2), (1, -1, -3), (4, -3, 1 ) respectively. 14. Show that area of a parallelogram having diagonals and is Show that ( x ) 2 = 16. Prove that the points A, B & C with position vectors respectively are collinear if and only if x + x + x =. 17. If x = x and x = x show that is parallel to, where, 18. Let be unit vectors such that. =. = 0 and the angle between is, prove that x 19. If are three vectors such that + + =, then prove that x = x = x. 20. If are vectors such that. =., x = x,, then show that. 21. Show that the vectors and are collinear. 22. Find, if ( ) x ( -λ + 7 ) =. 23. Let , and Find a vector which is is perpendicular to both and. = If are three vectors such that l l = 5, l l = 12 and l l = 13, and + + =, find the value of Find the position vector of a point R which divides the line joining the two points P and Q whose position vectors are (2 + and ( ) respectively, externally in the ratio 1 : 2. Also, show that P is the midpoint of the line segment RQ. 26. Find a unit vector perpendicular to each of the vectors + +2, If l l = 13, l l = 5 and. = 60, then find l x l. 28. If l l = 2, l l = 5 and l x l = 8, find If + +, are given vectors, then find a vector satisfying the equations x = and. = Express the vector as the sum of two vectors such that one is parallel to the vector + and other is perpendicular to. 31. If the vertices A, B, C of triangle ABC have position vectors (1, 2, 3) (-1, 0, 0), (0, 1, 2) respectively, what is the magnitude of angle ABC? 32. If is a unit vector, then find l l such that ( ). ( ) = 8.
7 33. Show that vector + + is equally inclined to the axes. 34. Show that three points, and are collinear. 35. If the sum of two unit vectors is a unit vector, show that the magnitude of their difference is. Answer Key 1. m = 2 3. ±1/, ±1/ ±1/ ; 8 ways 4. = 2 (± ± ± ) 5. 2/7, -3/7, 6/7 ; = = = = 8. i. p = -15 ii. p = 2/ ( ) ( ) ( , cos -1 ( ) 32. 3
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