2.
DEFINITION Origin : If X’OX, Y’OY, Z’OZ be three mutually perpendicular st. lines, which intersect at O, then O is called origin. Co-ordinate Axes : X’OX is called x-axis, its equations are y = 0, z = 0. Y’OY is called y-axis, its equations are x = 0, z = 0. Z’OZ is called z-axis, its equations are x = 0, y = 0. Co-ordinate Planes : The plane YOZ is called yz- plane, its equation is x = 0, the plane ZOX, the zx- plane, its equation is y = 0, The plane XOY, the xy-plane, its equation is z = 0.
3.
EXAMPLES1. Show that the points (0, 7, 10),(–1, 6, 6) and (– 4, 9, 6) form a right angled isosceles triangle.2. Show by using distance formula that the points (4, 5, –5), (0, –11, 3) and (2, –3, –1) are collinear.3. Find the locus of a point which moves such that the sum of its distances from points A(0, 0, – ) and B(0, 0, ) is constant. Ans :
4.
EXAMPLES1. One of the vertices of a cuboid is (1, 2, 3) and the edges from this vertex are along the +ve x-axis, +ve y- axis and +z axis respectively and are of length 2, 3, 2 respectively. Find out the vertices. Ans. (1, 2, 5), (3, 2, 5), (3, 2, 3), (1, – 1, 3), (1, – 1, 5), (3, – 1, 5), (3, – 1, 3).2. Show that the points (0, 4, 1), (2, 3, –1), (4, 5, 0) and (2, 6, 2) are the vertices of a square.3. Find the locus of point P if AP2 – BP2 = 18, where A (1, 2, – 3) and B (3, – 2, 1) Ans. 2x – 4y + 4z – 9 = 0
5.
DISTANCE BETWEEN TWO POINTS The distance between P (x1,y1,z1) and Q (x2,y2,z2) is 2 2 2 PQ x2 x1 y2 y1 z 2 z1
6.
SECTION FORMULA The position vector of a point R, which divides the segment joining P (x1, y1, z1) and Q (x2, y2, z2) in ratio m : mr2 n r1 n is r , m1 m2 where , 0 r1 , r2 position are m n vectors of the points P and Q respectively.
7.
DIRECTION-RATIOS AND DIRECTION-COSINES ˆ ˆ ˆ If P be (a, b, c) and OP r i.e. r a i b j c k , then <a, b, c> are called direction-ratios of r makes with x, y, z- axis respectively, then l cos , m cos , n cos are called direction-cosines of line OP . ˆ r.i a ˆ b ˆ c k .i i j ˆ ˆ a l cos r ˆ i r a 2 b 2 c2 b Similarly m cos a 2 b2 c2 c and n cos a 2 b2 c2
8.
DIRECTION-COSINES OF LINE JOINING TWOPOINTS Direction-cosines of PQ, where P (x1, y1, z1) and Q (x2, y2, z2) are x 2 x1 y 2 y1 z 2 z1 , , PQ PQ PQ
9.
EQUATIONS OF ST. LINE Equations of st. line passing through a given point A anda parallel to given vector r a. m is m Cartesian Form : Equations of st. line through A (x1, y1, z1) and parallel to line whose d.r. are a, b, c are x x1 y y1 z z1 a b c
10.
Two-Point Form : Equation of st. line through two given points A and B with position vector and respectively a b is r. a b a Cartesian Form : Equations of st. line through A (x1, y1, z1) and B (x2, y2, z2) are : x x1 y y1 z z1 x 2 x1 y 2 y1 z 2 z1
12.
ANGLE BETWEEN TWO LINES Angle between two lines r a b and r a1 b1 is : b.b1 b b1 cos sin b b1 b b1 Cartesian Form: Angle between two lines : x x1 y y1 z z1 and x x1 y y1 z z1 b1 b2 b3 b1 b2 b3 b1b1 b 2 b 2 b3 b3 is : cos 2 2 2 b1 b2 b3 b12 b22 b32 2 b 2 b3 b3 b 2 sin 2 b1 b12
13.
SHORTEST DISTANCE Shortest distance between two lines r1 a1 band 1 r2 a 2 b2 is : b1 b 2 . a 2 a1 d b1 b 2
14.
EXAMPLES1. Find the angle between the lines whose direction cosines are given by l + m + n = 0 and l 2 + m2 – n2 = 0. Ans. 60°2. P (6, 3, 2) , Q (5, 1, 4) , R (3, 3, 5) are vertices of a find Q. Ans. 90°3. Show that the direction cosines of a line which is perpendicular to the lines having directions cosines l1 m1 n1 and l2 m2 n2 respectively are proportional to m1n2 – m2n1 , n1l2 – n2l1 , l1m2 – l2m1
15.
EQUATIONS OF PLANE Standard Form of vector equation of plane is ˆ r .n d, ˆ where n is unit vector perpendicular to plane and d is the distance of the plane from the origin. Cartesian equation of plane is lx + my + nz = d, where l, m, n are direction-cosines of normal to the plane. Note : If is vector equation of r . A ˆ Bˆ C k i j ˆ D plane, then Ax + By + Cz = D is the cartesian equation of plane.
16.
Equation of plane passing a given point A aand ˆ perpendicular to a given direction nis ˆ r a .n . 0 Cartesian Form : Equation of the plane through A (x1, y1, z1) and perpendicular to a given direction with direction-ratios n1, n2, n3 is x x1 n1 y y1 n 2 z z1 n 3 0
17.
EXAMPLES Check whether these points are coplanar. If yes, find the equation of plane containing them A (1, 1, 1), B (0, – 1, 0), C (2, 1, –1), D (3, 3, 0) Ans. yes, 4x – 3y + 2z = 3 Find the plane passing through point (– 3, – 3, 1) and perpendicular to the line joining the points (2, 6, 1) and (1, 3, 0). Ans. x + 3y + z + 11 = 0 Find the equation of plane parallel to x + 5y – 4z + 5 = 0 and cutting intercepts on the axes whose rum is 150. Ans. x + 5y – 4z = 3000/19
19.
ANGLE BETWEEN TWO PLANES Angle between a1x b1y c1z d1 0 and a2x b2y c2z d2 0 is given by : a1a 2 b1b2 c1c2 cos 2 2 2 2 2 2 a1 b1 c1 a 2 b2 c2 a1 b1 c1 These two planes are parallel if and a2 b2 c2 perpendicular if a1a2 b1b2 c1c2 0.
20.
ANGLE BETWEEN TWO PLANES Angle between the line r a b and plane ˆ r .n dis b.nˆ sin . b Cartesian Form : The angle between the line x x1 y y1 z z1 and the plane ax + by + cz = d is l m n la mb nc sin l2 m 2 n 2 a 2 b 2 c2
21.
DISTANCE OF A POINT FROM A PLANE Distance of point P from the plane ˆ r .n d is ˆ .n d . Cartesian Form : Perpendicular distance from P (x1, y1, z1) to the plane lx my nz d 0 is p lx1 my1 nz1 d.
22.
PLANE THROUGH THE INTERSECTION OF TWOPLANES Equation of any plane passing through the intersection of ˆ and r .n1 d1 ˆ is : r .n 2 d 2 ˆ r . n1 k n 2 d 1 kd1
23.
INTERCEPT FORM Equation of the plane in terms of a, b, c ; the x y z intercepts, of the plane on axes, is 1 a b c
24.
PLANE THROUGH A LINE x x1 y y1 z z1 is the equation of a st. line through ( l m n x1 , y1 , z1 ) with d. c. , l,m,n Equation of any plane through this line is : A (x x1) B (y y1) C (z z1) 0,where Al Bm Cn 0
25.
LENGTH OF PROJECTION Length of projection of AB upon CD with d. c. l, m, n is : (x2 x1)l (y2 y1)m (z2 z1)n , where A is (x1,y1,z1) and B is (x2,y2,z2).
26.
THREE POINT FORM Equation of the plane through (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) is : x y z 1 x1 y1 z1 1 0 x2 y2 z2 1 x3 y3 z3 1
27.
INTERCEPTS Intercepts of plane ˆ r .n p on axes are p p p ,ˆ , ˆ ˆ .n j.n k .n i
28.
ANGLE BETWEEN TWO PLANES ˆ Angle between planes r .n ˆ pand r .n p is n .n cos n n Planes ˆ ˆ r .n p and r .n p are parallel if n n and perpendicular if . n.n 0
29.
LENGTH OF PERPENDICULAR Length of perpendicular from a point P r1upon the line is : r a b a r1 b b
30.
BISECTORS Bisectors of the angles between two st. lines r a b and r a c are r a t c b and r a t c b .
31.
EXAMPLES A tetrahedron has vertices at O(0, 0, 0), A(1, 2, 1), B(2, 1, 3) and C(– 1, 1, 2). Prove that the angle between the faces OAB and ABC will be cos–1 (19/35). Find the equation of plane passing through the line of intersection of the planes 4x – 5y – 4z = 1 and 2x + y + 2z = 8 and the point (2, 1, 3). Ans. 32x – 5y + 8z – 83 = 0, = 10/3 Find the equations of the planes bisecting the angles between the planes x + 2y + 2z – 3 = 0, 3x + 4y + 12z + 1 = 0 and specify the plane which bisects the acute angle between them. Ans. 2x + 7y – 5z = 21 ; 11x + 19y + 31z = 18 ; 2x + 7y – 5z = 21 Show that the origin lies in the acute angle between the planes x + 2y + 2z – 9 = 0 and 4x – 3y + 12z + 13 = 0 Prove that the planes 12x – 15y + 16z – 28 = 0, 6x + 6y – 7z – 8 = 0 and 2x + 35y – 39z + 12 = 0 have a common line of intersection.