THREE DIMENSIONAL GEOMETRY. The moving power of mathematical invention is not reasoning but imagination. A. DEMORGAN

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1 THREE D IMENSIONAL G EOMETRY 463 The moving powe of mathematical invention is not easoning but imagination. A. DEMORGAN. Intoduction In Class XI, while studying Analytical Geomety in two dimensions, and the intoduction to thee dimensional geomety, we confined to the Catesian methods only. In the pevious chapte of this book, we have studied some basic concepts of vectos. We will now use vecto algeba to thee dimensional geomety. T he pupose of this appoach to 3-dimensional geomety is that it makes the study simple and elegant*. In this chapte, we shall study the diection cosines and diection atios of a line joining two points and also discuss about the equations of lines and planes in space unde diffeent conditions, angle between two lines, two Leonhad Eule planes, a line and a plane, shotest distance between two ( skew lines and distance of a point fom a plane. Most of the above esults ae obtained in vecto fom. Nevetheless, we shall also tanslate these esults in the Catesian fom which, at times, pesents a moe clea geometic and analytic pictue of the situation.. Diection Cosines and Diection Ratios of a Line Fom Chapte 0, ecall that if a diected line L passing though the oigin makes angles α, β and γ with x, y and z-axes, espectively, called diection angles, then cosine of these angles, namely, cos α, cos β and cos γ ae called diection cosines of the diected line L. If we evese the diection of L, then the diection angles ae eplaced by thei supplements,. Thus, the signs of the diection cosines ae evesed. i.e., π α, π βand π γ Chapte THREE DIMENSIONAL GEOMETRY not to be epublished * Fo vai ou s acti vi ti es i n th ee di mens io nal geomet y, on e may efe to t he Boo k A Hand Book fo designing Mathematics Laboatoy in Schools, NCERT, 005

2 464 MATHE MATI CS Fig. Note that a given line in space can be extended in two opposite diections and so it has two sets of diection cosines. In ode to have a unique set of diection cosines fo a given line in space, we must take the given line as a diected line. These unique diection cosines ae denoted by l, m and n. Remak If the given line in space does not pass though the oigin, then, in ode to find its diection cosines, we daw a line though the oigin and paallel to the given line. Now take one of the diected lines fom the oigin and find its diection cosines as two paallel line have same set of diection cosines. Any thee numbes which ae popotional to the diection cosines of a line ae called the diection atios of the line. If l, m, n ae diection cosines and a, b, c ae diection atios of a line, then a λl, bλm and c λn, fo any nonzeo λ R. Note Some authos also call diection atios as diection numbes. Let a, b, c be diection atios of a line and let l, m and n be the diection cosines (d.c s of the line. Then l a m b n k c (say, k being a constant. Theefoe l ak, m bk, n ck... ( But l + m + n Theefoe k (a + b + c not to be epublished o k ± a + b + c

3 THREE D IMENSIONAL G EOMETRY 465 Hence, fom (, the d.c. s of the line ae a b c l ±, m ±, n ± a + b + c a + b + c a + b + c whee, depending on the desied sign of k, eithe a positive o a negative sign is to be taken fo l, m and n. Fo any line, if a, b, c ae diection atios of a line, then ka, kb, kc; k 0 is also a set of diection atios. So, any two sets of diection atios of a line ae also popotional. Also, fo any line thee ae infinitely many sets of diection atios... Relation between the diection cosines of a line Conside a line RS with diection cosines l, m, n. Though the oigin daw a line paallel to the given line and take a point P(x, y, z on this line. Fom P daw a pependicula PA on the x-axis (Fig... OA x Let OP. Then cosα. This gives x l. OP Similaly, y m and z n Thus x + y + z (l + m + n But x + y + z Hence l + m + n.. Diection cosines of a line passing though two points Since one and only one line passes though two given points, we can detemine the diection cosines of a line passing though the given points P(x, y, z and Q(x, y, z as follows (Fig.3 (a. Fig.3 X A R Z O α O Fig. S P (,, not to be epublished α x x y z Y P A

4 466 MATHE MATI CS Let l, m, n be the diection cosines of the line PQ and let it makes angles α, β and γ with the x, y and z-axis, espectively. Daw pependiculas fom P and Q to XY-plane to meet at R and S. Daw a pependicula fom P to QS to meet at N. Now, in ight angle tiangle PNQ, PQN γ (Fig.3 (b. T heefoe, cosγ NQ PQ z z PQ x Similaly cosα x y and cos y β PQ PQ Hence, the diection cosines of the line segment joining the points P(x, y, z and Q(x, y, z ae x PQ x, y y z, PQ whe e PQ ( x x ( y y ( z z PQ + + Note The diection atios of the line segment joining P(x, y, z and Q(x, y, z may be taken as x x, y y, z z o x x, y y, z z Example If a line makes angle 90, 60 and 30 with the positive diection of x, y and z-axis espectively, find its diection cosines. Solution Let the d.c.'s of the lines be l, m, n. Then l cos , m cos 60 0, 3 n cos Example If a line has diection atios,,, detemine its diection cosines. Solution Diection cosines ae + ( + (, + ( + (, not to be epublished z + ( + ( o,, Example 3 Find the diection cosines of the line passing though the two points (, 4, 5 and (,, 3.

5 THREE D IMENSIONAL G EOMETRY 467 Solution We know the diection cosines of the line passing though two points P(x, y, z and Q(x, y, z ae given by x x y y z z,, PQ PQ PQ whee PQ ( Hee P is (, 4, 5 and Q is (,, 3. ( x + z z x + ( y y So PQ ( ( + ( 4 + (3 ( 5 77 T hus, the diection cosines of the line joining two points is 3,, Example 4 Find the diection cosines of x, y and z-axis. Solution T he x-axis makes angles 0, 90 and 90 espectively with x, y and z-axis. T heefoe, the diection cosines of x-axis ae cos 0, cos 90, cos 90 i.e.,,0,0. Similaly, diection cosines of y-axis and z-axis ae 0,, 0 and 0, 0, espectively. Example 5 Show that the points A (, 3, 4, B (,, 3 and C (3, 8, ae collinea. Solution Diection atios of line joining A and B ae, 3, i.e.,, 5, 7. T he diection atios of line joining B and C ae 3, 8 +, 3, i.e.,, 0, 4. It is clea that diection atios of AB and BC ae popotional, hence, AB is paallel to BC. But po int B is common to both AB and BC. Theefoe, A, B, C ae collinea points. EXERCIS E.. If a line makes angles 90, 35, 45 with the x, y and z-axes espectively, find its diection cosines.. Find the diection cosines of a line which makes equal angles with the coodinate axes. not to be epublished 3. If a line has the diection atios 8,, 4, then what ae its diection cosines? 4. Show that the points (, 3, 4, (,,, (5, 8, 7 ae collinea. 5. Find the diection cosines of the sides of the tiangle whose vetices ae (3, 5, 4, (,, and ( 5, 5,.

6 468 MATHE MATI CS.3 Equation of a Line in Space We have studied equation of lines in two dimensions in Class XI, we shall now study the vecto and catesian equations of a line in space. (i (ii A line is uniquely detemined if it passes though a given point and has given diection, o it passes though two given points..3. Equation of a line though a given point and paallel to a given vecto b Let a be the position vecto of the given point A wit h esp ect to th e o igin O o f t he ectangula coodinate system. Let l be the line which passes though the point A and is paallel to a given vecto b. Let be the position vecto of an abitay point P on the line (Fig.4. uuu Then AP is paallel to the vecto b, i.e., uuu AP λb, whee λ is some eal numbe. uuu uuu uuu But AP OP OA i.e. λb a Convesely, fo each value of the paamete λ, this equation gives the position vecto of a point P on the line. Hence, the vecto equation of the line is given by a + λ b... ( Remak If b ai + bj + ck, then a, b, c ae diection atios of the line and convesely, if a, b, c ae diection atios of a line, then b ai + bj + ck will be the paallel to the line. Hee, b should not be confused with b. Deivation of cate sian fom fom ve cto fom Fig.4 Let the coodinates of the given point A be (x, y, z and the diection atios of the line be a, b, c. Conside the coodinates of any point P be (x, y, z. Then xi + yj + zk ; a x i y j z k + + and b a i + b j + c k Substituting these values in ( and equating the coefficients of i, j and k, we get x x + λa; y y + λ b; z z + λc... ( not to be epublished

7 THREE D IMENSIONAL G EOMETRY 469 T hese ae paametic equations of the line. Eliminating the paamete λ fom (, we get x x y y z z a b c T his is the Catesian equation of the line.... (3 Note If l, m, n ae the diection cosines of the line, the equation of the line is x x l y y m z z n Example 6 Find the vecto and the Catesian equations of the line though the point (5,, 4 and which is paallel to the vecto 3 i + j 8k. Solution We have a 5 i + j 4 k and b 3i + j 8k Theefoe, the vecto equation of the line is 5 i + j 4 k + λ ( 3 i + j 8 k Now, is the position vecto of any point P(x, y, z on the line. Theefoe, xi + y j + z k 5 i + j 4 k + λ ( 3 i + j 8 k Eliminating λ, we get (5 + 3 λ i$ + ( + λ $ j + ( 4 8 λ k$ x 5 y z which is the equation of the line in Catesian fom..3. Equation of a line passing though two given points Let a and b be the position vectos of two po in ts A (x, y, z an d B(x, y, z, espectively that ae lying on a line (Fig.5. Let be t he positio n vecto of an abitay point P(x, y, z, then P is a point on uuu th e lin e if and on ly if AP a an d uuu AB b a ae collinea vectos. Theefoe, P is on the line if and only if a ( b a not to be epublished λ Fig.5

8 470 MATHE MATI CS o a + λ( b a, λ R.... ( This is the vecto equation of the line. Deivation of cate sian fom fom ve cto fom We have xi y j z k, a x i y j z k and b x i y j z k, + + Substituting these values in (, we get xi $ + y $ j + z k$ x $ i + y $ j + z k$ + λ [( x x $ i + ( y y $ j + ( z z k$ ] Equating the like coefficients of i, j, k, we get x x + λ (x x ; y y + λ (y y ; z z + λ (z z On eliminating λ, we obtain x x y y z z x x y y z z which is the equation of the line in Catesian fom. Example 7 Find the vecto equation fo the line passing though the points (, 0, and (3, 4, 6. Solution Let a and b be the position vectos of the point A(, 0, and B(3, 4, 6. Then and Theefoe + a i k b 3i + 4 j + 6 k b a 4 i + 4 j + 4 k Let be the position vecto of any point on the line. Then the vecto equation of the line is i k (4 i 4 j 4 k + +λ + + Example 8 The Catesian equation of a line is + + x 3 y 5 z 6 4 Find the vecto equation fo the line. not to be epublished Solution Compaing the given equation with the standad fom x x y y z z a b c We obseve that x 3, y 5, z 6; a, b 4, c.

9 THREE D IMENSIONAL G EOMETRY 47 vecto Thus, the equied line passes though the point ( 3, 5, 6 and is paallel to the i + 4 j + k. Let be the position vecto of any point on the line, then the vecto equation of the line is given by + ( 3 i 5 j 6 k + λ (i + 4 j + k.4 Angle between Two Lines Let L and L be two lines passing though the oigin and with diection atios a, b, c and a, b, c, espectively. Let P be a point on L and Q be a point on L. Conside the diected lines OP and OQ as given in Fig.6. Let θ be the acute angle between OP an d OQ. No w ecall that the diected line segments OP and OQ ae vectos with components a, b, c and a, b, c, espectively. Theefoe, the angle θ between them is given by cos θ a a + b b + c c a b c a b c The angle between the lines in tems of sin θ is given by sin θ cos θ ( a a + b b + c c ( a + b + c ( a + b + c ( a + b + c ( a + b + c ( aa + b b + cc ( a + b + c ( a + b + c Fig a + b + c a + b + c ( a b a b ( b c b c ( c a c a... (... ( not to be epublished Note In case the lines L and L do not pass though the oigin, we may take lines L and L which ae paallel to L and L espectively and pass though the oigin.

10 47 MATHE MATI CS If instead of diection atios fo the lines L and L, diection cosines, namely, l, m, n fo L and l, m, n fo L ae given, then ( and ( takes the following fom: cos θ l l + m m + n n (as l + m + n and sin θ ( l + m + n... (3 l m l m ( m n m n + ( n l n l... (4 Two lines with diection atios a, b, c and a, b, c ae (i pependicula i.e. if θ 90 by ( (ii paallel i.e. if θ 0 by ( a a + b b + c c 0 a a b c b c Now, we find the angle between two lines when thei equations ae given. If θ is acute the angle between the lines a +λ u b and a +µ b b b then cosθ b b In Catesian fom, if θ is the angle between the lines and x x a x x a y y z z... ( b c y y z z... ( b c whee, a, b, c and a, b, c ae the diection atios of the lines ( and (, espectively, then cos θ a a + b b + c c not to be epublished a b c a b c Example 9 Find the angle between the pai of lines given by 3 i + j 4 k + λ ( i + j + k and 5 i j + µ (3 i + j + 6 k

11 THREE D IMENSIONAL G EOMETRY 473 Solution Hee b i + j + k and b 3i + j + 6k The angle θ between the two lines is given by b b ( i + j + k (3i + j + 6 k cos θ b b Hence θ cos Example 0 Find the angle between the pai of lines x + 3 y z x + y 4 z 5 and Solution The diection atios of the fist line ae 3, 5, 4 and the diection atios of the second line ae,,. If θ is the angle between them, then cos θ Hence, the equied angle is cos 5..5 S hotest Distance between Two Lines If two lines in space intesect at a point, then the shotest distance between them is zeo. Also, if two lines in space ae paallel, then the shotest distance between them will be the pependicula distance, i.e. the length of the pependicula dawn fom a point on one line onto the othe line. not to be epublished Futhe, in a space, thee ae lines which ae neithe intesecting no paallel. In fact, such pai of lines ae non coplana and ae called skew lines. Fo example, let us conside a oom of size, 3, units along x, y and z-axes espectively Fig.7. Fig.7

12 474 MATHE MATI CS T he line GE that goes diagonally acoss the ceiling and the line DB passes though one cone of the ceiling diectly above A and goes diagonally down the wall. These lines ae skew because they ae not paallel and also neve meet. By the shotest distance between two lines we mean the join of a point in one line with one point on the othe line so that the length of the segment so obtained is the smallest. Fo skew lines, the line of the shotest distance will be pependicula to both the lines..5. Distance between two skew lines We now detemine the shotest distance between two skew lines in the following way: Let l and l be two skew lines with equations (Fig..8 a + λ b... ( and a + µ b... ( Take any point S on l with position vecto a and T on l, with position vecto a. T hen the magnitude of the shotest distance vecto T will be equal to that of the pojection of ST along the Q diection of the line of shotest distance (See l If PQ uuu is the shotest distance vecto between l and l, then it being pependicula to both b and b, the unit vecto n along PQ would theefoe be S P l Fig.8 n b b Then PQ d n b b... (3 whee, d is the magnitude of the shotest distance vecto. Let θ be the angle between ST and PQ. Then PQ ST cos θ uuu uu PQ ST But cos θ uuuu uu PQ ST d n ( a a uu (since ST a d ST a ( b b ( a a ST b b [Fom (3] not to be epublished

13 THREE D IMENSIONAL G EOMETRY 475 Hence, the equied shotest distance is d PQ ST cos θ o d ( b b. ( a a b b Cate sian fom T he shotest distance between the lines l : x x a x x and l : a is.5. Distance between paallel lines y y z z b c y y z z b c x x y y z z a b c a b c + + ( b c b c ( c a c a ( a b a b If two lines l and l ae paallel, then they ae coplana. Let the lines be given by a + λ b... ( and a + µ b ( whee, a is the position vecto of a point S on l and a is the position vecto of a point T on l Fig.9. As l, l ae coplana, if the foot of the pependicula fom T on the line l is P, then the distance between the lines l and l TP. Let θ be the angle between the vectos ST uu and b. Then b ST uu uu ( b ST sin θ n whee n is the unit vecto pependicula to the plane of the lines l and l. But ST uu a a Fig.9 not to be epublished... (3

14 476 MATHE MATI CS Theefoe, fom (3, we get b ( a a b PT n (since PT ST sin θ i.e., b ( a a b PT (as n Hence, the distance between the given paallel lines is uuu b ( a a d PT b Example Find the shotest distance between the lines l and l whose vecto equations ae i + j + λ ( i j + k... ( and i + j k + µ (3 i 5 j + k... ( Solution Compaing ( and ( with a + λ b and a + µ b espectively, we get a i + j, b i j + k a i + j k and b 3 i 5 j + k T heefoe a a i k and b b ( i j + k ( 3 i 5 j + k So b b i j k 3 i j 7 k Hence, the shotest distance between the given lines is given by ( b b. ( a a d b b not to be epublished Example Find the distance between the lines l and l given by i + j 4 k + λ ( i + 3 j + 6 k and 3 i + 3 j 5 k + µ ( i + 3 j + 6 k

15 THREE D IMENSIONAL G EOMETRY 477 Solution The two lines ae paallel (Why? We have a i + j 4k, a 3 i + 3 j 5k and b i + 3 j + 6k T heefoe, the distance between the lines is given by d o b ( a a b i j k i + 4 j 4 k EXERCIS E.. Show that the thee lines with diection cosines ,, ;,, ;,, ae mutually pependicula Show that the line though the points (,,, (3, 4, is pependicula to the line though the points (0, 3, and (3, 5, Show that the line though the points (4, 7, 8, (, 3, 4 is paallel to the line though the points (,,, (,, Find the equation of the line which passes though the point (,, 3 and is paallel to the vecto 3 i + j k. 5. Find the equation of the line in vecto and in catesian fom that passes though the point with position vecto i j + 4 k and is in the diection i + j k. 6. Find the catesian equation of the line which passes though the point (, 4, 5 and paallel to the line given by x + 3 y 4 z not to be epublished 7. x 5 y + 4 z 6 The catesian equation of a line is. Wite its vecto fom Find the vecto and the catesian equations of the lines that passes though the oigin and (5,, 3.

16 478 MATHE MATI CS 9. Find the vecto and the catesian equations of the line that passes though the points (3,, 5, (3,, Find the angle between the following pais of lines: (i i 5 j + k + λ (3 i + j + 6 k and 7 i 6 k + µ ( i + j + k (ii 3i + j k + λ ( i j k and i j 56 k + µ (3 i 5 j 4 k. Find the angle between the following pai of lines: (i x y z and 5 x 3 y z 8 4 x y z x 5 y z 3 (ii and 4 8. Find the values of p so that the lines x 7 y 4 z 3 3 p and 7 7 x y 5 6 z ae at ight angles. 3 p 5 x 5 y + z x y z 3. Show that the lines and ae pependicula to each othe. 4. Find the shotest distance between the lines ( i + j + k + λ ( i j + k and i j k + µ ( i + j + k 5. Find the shotest distance between the lines x + y + z + x 3 y 5 z 7 and Find the shotest distance between the lines whose vecto equations ae ( i + j + 3 k + λ ( i 3 j + k and 4 i + 5 j + 6 k + µ ( i + 3 j + k 7. Find the shotest distance between the lines whose vecto equations ae ( t i + ( t j + (3 t k and ( s + i + (s j (s + k not to be epublished

17 THREE D IMENSIONAL G EOMETRY Plane A plane is detemined uniquely if any one of the following is known: (i (ii (iii the nomal to the plane and its distance fom the oigin is given, i.e., equation of a plane in nomal fom. it passes though a point and is pependicula to a given diection. it passes though thee given non collinea points. Now we shall find vecto and Catesian equations of the planes..6. Equation of a plane in nomal fom Conside a plane whose pependicula distance fom the oigin is d (d 0. Fig.0. uuu If ON is the nomal fom the oigin to the plane, and n is the unit nomal vecto uuu uuu along ON. Then ON d n Z. Let P be any uuu po int on th e p lan e. T heefo e, NP is uuu pependicula to ON. uuu uuu T heefoe, NP ON 0... ( Let be the position vecto of the point P, uuu uuu uuu uuu then NP d n (as ON NP OP T heefoe, ( becomes + ( d n d n 0 o ( d n n 0 (d 0 o n d n n 0 i.e., n d (as n n ( This is the vecto fom of the equation of the plane. Cate sian fom Equation ( gives the vecto equation of a plane, whee n is the unit vecto nomal to the plane. Let P(x, y, z be any point on the plane. Then uuu OP x i y j z k + + Let l, m, n be the diection cosines of n. Then n + + l i m j n k X O P( x, y,z d N Fig.0 not to be epublished Y

18 480 MATHE MATI CS T heefoe, ( gives ( x i + y j + z k ( l i + m j + n k d i.e., lx + my + nz d... (3 This is the catesian equation of the plane in the nomal fom. Note Equation (3 shows that if ( a i + b j + c k d is the vecto equation of a plane, then ax + by + cz d is the Catesian equation of the plane, whee a, b and c ae the diection atios of the nomal to the plane. Example 3 Find the vecto equation of the plane which is at a distance of fom the oigin and its nomal vecto fom the oigin is i 3 j 4k Solution Let n i 3 j + 4 k. Then n i 3 j + 4 k i 3 j + 4 k n n Hence, the equied equation of the plane is i + j + k Example 4 Find the diection cosines of the unit vecto pependicula to the plane (6 i 3 j k + 0 passing though the oigin. Solution The given equation can be witten as ( 6 i + 3 j + k... ( + + No w 6 i + 3 j + k Theefoe, dividing both sides of ( by 7, we get 6 3 i + j + k which is the equation of the plane in the fom n This shows that d. not to be epublished 6 3 n i + j + k is a unit vecto pependicula to the plane though the oigin. Hence, the diection cosines of n ae 6 3,,

19 THREE D IMENSIONAL G EOMETRY 48 Example 5 Find the distance of the plane x 3y + 4z 6 0 fom the oigin. Solution Since the diection atios of the nomal to the plane ae, 3, 4; the diection cosines of it ae 3 4,, + ( ( ( X, i.e., O Z 3 4,, Hence, dividing the equation x 3y + 4z 6 0 i.e., x 3y + 4z 6 thoughout by 9, we get x + y + z This is of the fom lx + my + nz d, whee d is the distance of the plane fom the oigin. So, the distance of the plane fom the oigin is 6. 9 Example 6 Find the coodinates of the foot of the pependicula dawn fom the oigin to the plane x 3y + 4z 6 0. Solution Let the coodinates of the foot of the pependicula P fom the oigin to the plane is(x, y, z (Fig.. Then, the diection atios of the line OP ae x, y, z. Witing the equation of the plane in the nomal fom, we have x y + z wh ee,,, cosines of the OP. ae th e diect io n Since d.c. s and diection atios of a line ae popotional, we have x 9 i.e., x y z k k, 9 y 3k 4k, z 9 9 Fig. P(x, y, z not to be epublished Y

20 48 MATHE MATI CS Substituting these in the equation of the plane, we get k 8 4 Hence, the foot of the pependicula is,, Note If d is the distance fom the oigin and l, m, n ae the diection cosines of the nomal to the plane though the oigin, then the foot of the pependicula is (ld, md, nd..6. Equation of a plane pependicula to a given vecto and passing though a given point In the space, thee can be many planes that ae pependicula to the given vecto, but though a given point P(x, y, z, only one such plane exists (see Fig.. Let a plane pass though a point A with position vecto a and pependicula to the vecto N u. Let be the position vecto of any point P(x, y, z in the plane. (Fig.3. Then the point P lies in the plane if and only if uuu u uuu u AP is pependicula to N. i.e., AP. N 0. But uuu AP a. Theefoe, ( a N 0 ( This is the vecto equation of the plane. Cate sian fom Let the given point A be (x, y, z, P be (x, y, z and diection atios of N u ae A, B and C. Then, Fig.3 a x i + y j+ z k, xi + y j + z k and N Ai + B j + Ck No w ( a N 0 x x i + y y j + z z k (Ai + B j + C k 0 So ( ( ( i.e. A (x x + B (y y + C (z z 0 Fig. not to be epublished Example 7 Find the vecto and catesian equations of the plane which passes though the point (5,, 4 and pependicula to the line with diection atios, 3,.

21 X O Z P R a S b c THREE D IMENSIONAL G EOMETRY 483 Solution We have the position vecto of point (5,, 4 as a 5i + j 4k and the nomal vecto N pependicula to the plane as N i + 3 j k Theefoe, the vecto equation of the plane is given by ( a.n 0 o [ (5 i + j 4 k ] ( i + 3 j k 0... ( T ansfoming ( into Catesian fom, we have [( x 5 i + ( y j + ( z + 4 k ] ( i + 3 j k o ( x 5 3( y ( z 4 0 i.e. x + 3y z 0 which is the catesian equation of the plane..6.3 Equation of a plane passing though thee non collinea points Let R, S and T be thee non collinea points on the plane with position vectos a,b and c espectively (Fig.4. (RS X RT Fig.4 uuu uuu uuu The vectos RS and RT ae in the given plane. Theefoe, the vecto RS is pependicula to the plane containing points R, S and T. Let be the position vecto of any point P in the plane. Theefoe, the equation of the plane passing though R and uuu uuu pependicula to the vecto RS RT is uuu uuu ( a (RS RT 0 o ( a.[( b a ( c a ] 0 ( T not to be epublished Y uuu RT

22 484 MATHE MATI CS T his is the equation of the plane in vecto fom passing though thee noncollinea points. Note Why was it necessay to say that the thee points had to be non collinea? If the thee points wee on the same line, then thee will be many planes that will contain them (Fig.5. T hese planes will esemble the pages of a book whee the T line containing the points R, S and T ae membes in the binding of the book. Cate sian fom Fig.5 Let (x, y, z, (x, y, z and (x 3, y 3, z 3 be the coodinates of the points R, S and T espectively. Let (x, y, z be the coodinates of any point P on the plane with position vecto. Then uuu RP (x x i + (y y j + (z z k uuu RS (x x i + (y y j + (z z k uuu RT (x3 x i + (y 3 y j + (z 3 z k Substituting these values in equation ( of the vecto fom and expessing it in the fom of a deteminant, we have x x y y z z x x y y z z 0 x x y y z z which is the equation of the plane in Catesian fom passing though thee non collinea points (x, y, z, (x, y, z and (x 3, y 3, z 3. Example 8 Find the vecto equations of the plane passing though the points R(, 5, 3, S(, 3, 5 and T(5, 3, 3. Solution Let a i 5 j 3k, b i 3 j + 5 k, c 5i 3 j 3k + + Then the vecto equation of the plane passing though a, b and c and is given by uuu uuu ( a (RS RT 0 (Why? o ( a [( b a ( c a] 0 not to be epublished + + i.e. [ ( i 5 j 3 k ] [( 4i 8 j 8 k (3i j ] 0 R S

23 THREE D IMENSIONAL G EOMETRY Intecept fom of the equation of a plane In this section, we shall deduce the equation of a plane in tems of the intecepts made by the plane on the coodinate axes. Let the equation of the plane be Ax + By + Cz + D 0 (D 0... ( Let the plane make intecepts a, b, c on x, y and z axes, espectively (Fig.6. Hence, the plane meets x, y and z-axes at (a, 0, 0, (0, b, 0, (0, 0, c, espectively. Theefoe Aa + D 0 o A Bb + D 0 o B D a D b D Cc + D 0 o C c Substituting these values in the equation ( of the plane and simplifying, we get x y z ( a b c which is the equied equation of the plane in the intecept fom. Example 9 Find the equation of the plane with intecepts, 3 and 4 on the x, y and z-axis espectively. Solution Let the equation of the plane be x y z ( a b c Hee a, b 3, c 4. Substituting the values of a, b and c in (, we get the equied equation of the x y z plane as + + o 6x + 4y + 3z Plane passing tho ugh the intesection of two given planes Let π an d π be two p lan es with equat io ns n d and n d espectively. The position vecto of any point on the line of intesection must satisfy both the equations (Fig.7. Fig.6 not to be epublished Fig.7

24 486 MATHE MATI CS If t is the position vecto of a point on the line, then t n d and t n d Theefoe, fo all eal values of λ, we have t ( n +λn d +λd Since t is abitay, it satisfies fo any point on the line. Hence, the equation ( n + λ n d +λd epesents a plane π 3 which is such that if any vecto satisfies both the equations π and π, it also satisfies the equation π 3 i.e., any plane passing though the intesection of the planes n d and n d has the equation ( n + λn d + λd... ( Cate sian fom In Catesian system, let n A i + B j+ C k n A i + B j + C k and xi + y j + z k Then ( becomes x (A + λa + y (B + λb + z (C + λc d + λd o (A x + B y + C z d + λ(a x + B y + C z d 0... ( which is the equied Catesian fom of the equation of the plane passing though the intesection of the given planes fo each value of λ. Example 0 Find the vecto equation of the plane passing though the intesection of the planes ( i + j + k 6 and (i + 3 j + 4 k 5, and the point (,,. Solution Hee, n i + j + k and n i + 3 j + 4 k ; and d 6 and d 5 Hence, using the elation ( n + λ n d + λ d, we get not to be epublished o [ i + j + k +λ (i + 3 j + 4 k ] 6 5λ [(+ λ i + ( + 3 λ j + (+ 4 λ k ] 6 5λ ( whee, λ is some eal numbe.

25 Taking o + + xi y j z k, we get THREE D IMENSIONAL G EOMETRY 487 ( xi + y j + z k [( + λ i + (+ 3 λ j + (+ 4 λ k ] 6 5λ ( + λ x + ( + 3λ y + ( + 4λ z 6 5λ o (x + y + z 6 + λ (x + 3y + 4 z ( Given that the plane passes though the point (,,, it must satisfy (, i.e. o λ ( λ ( Putting the values of λ in (, we get o o i j k i + j + k (0i + 3 j + 6 k 69 which is the equied vecto equation of the plane..7 Coplanaity of Two Lines Let the given lines be a +λb and a + µ b (... ( T he line ( passes though the point, say A, with position vecto a and is paallel to b. The line ( passes though the point, say B with position vecto a and is paallel to b. uuu Thus, AB a a uuu The given lines ae coplana if and only if AB is pependicula to b b. uuu i.e. AB.( b b 0 o ( a a ( b b 0 Cate sian fom Let (x, y, z and (x, y, z be the coodinates of the points A and B espectively. not to be epublished

26 488 MATHE MATI CS Let a, b, c and a, b, c be the diection atios of b and b, espectively. Then uuu AB ( x x i + ( y y j+ ( z z k b a i + b j+ c k and b a i + b j+ c k uuu The given lines ae coplana if and only if AB ( b b 0. In the catesian fom, it can be expessed as... (4 x x y y z z a b c a b c Example Show that the lines x +3 y z 5 x + y z 5 and ae coplana Solution Hee,x 3, y, z 5, a 3, b, c 5 x, y, z 5, a, b, c 5 Now, conside the deteminant T heefoe, lines ae coplana. 0 a b c 0 x x y y z z a b c Angle between Two Planes De finition T he angle between two planes is defined as the angle between thei nomals (Fig.8 (a. Obseve that if θ is an angle between the two planes, then so is 80 θ (Fig.8 (b. We shall take the acute angle as the angles between two planes. not to be epublished Fig.8

27 THREE D IMENSIONAL G EOMETRY 489 If n and n ae nomals to the planes and θ be the angle between the planes n d and. n d. Then θ is the angle between the nomals to the planes dawn fom some common point. n n We have, cos θ n n Note The planes ae pependicula to each othe if n. n 0 and paallel if n is paallel to n. Catesian fom Let θ be the angle between the planes, A x + B y + C z + D 0 and A x + B y + C z + D 0 The diection atios of the nomal to the planes ae A, B, C and A, B, C espectively. Theefoe, cos θ A A + B B + C C A B C A B C Note. If th e p lan es ae at ight angles, t hen θ 9 0 o and so cos θ 0. Hence, cos θ A A + B B + C C 0.. A B C. If the planes ae paallel, then A B C Example Find the angle between the two planes x + y z 5 and 3x 6y z 7 using vecto method. Solution The angle between two planes is the angle between thei nomals. Fom the equation of the planes, the nomal vectos ae N u i + j k u and N 3 i 6 j k u u N N (i + j k (3 i 6 j k 4 Theefoe cos θ u u N N not to be epublished 4 Hence θ cos

28 490 MATHE MATI CS Example 3 Find the angle between the two planes 3x 6y + z 7 and x + y z 5. Solution Compaing the given equations of the planes with the equations A x + B y + C z + D 0 and A x + B y + C z + D 0 We get A 3, B 6, C A, B, C cos θ 5 3 Theefoe, θ cos ( 6 ( + ( ( ( 3 + ( 6 + ( ( + + ( Distance of a Point fom a Plane Vecto fom Conside a point P with positio n vecto a and a plane π whose equation is n d (Fig.9. Z Z X O a P N π (a d Q N π Fig.9 Conside a plane π though P paallel to the plane π. The unit vecto nomal to π is n. Hence, its equation is ( a n 0 i.e., n a n Thus, the distance ON of this plane fom the oigin is a n. T heefoe, the distance PQ fom the plane π is (Fig.. (a i.e., ON ON d a n π Y X N (b P d N O not to be epublished a π Y

29 THREE D IMENSIONAL G EOMETRY 49 which is the length of the pependicula fom a point to the given plane. We may establish the simila esults fo (Fig.9 (b. Note. If the equationof the plane π is in the fom u N d, whee N u is nomal to the plane, then the pependicula distance is a N u d u. N. T he length of the pependicula fom oigin O to the plane u N d is d u N Cate sian fom (since a 0. Let P(x, y, z be the given point with position vecto a and Ax + By + Cz D be the Catesian equation of the given plane. Then a x i + y j + z k u N A i + B j + Ck Hence, fom Note, the pependicula fom P to the plane is ( x i + y j + z k ( A i + B j + C k D A + B + C + + A + B + C A x B y C z D Example 4 Find the distance of a point (, 5, 3 fom the plane ( 6 i 3 j + k 4 u Solution Hee, a i 5 j 3 k, N 6 i 3 j k and d 4. not to be epublished + + Theefoe, the distance of the point (, 5, 3 fom the given plane is ( i + 5 j 3 k (6 i 3 j + k i 3 j + k

30 49 MATHE MATI CS.0 Angle between a Line and a Plane Definition 3 The angle between a line and a plane is the complement of the angle between the line and nomal to the plane (Fig.0. Vecto fom If th e equation of the line is a + λ b a nd the equation of the pla ne is n d. Then the angle θ between the line and the nomal to the plane is b n cos θ b n and so the angle φ between the line and the plane is given by 90 θ, i.e., sin (90 θ cos θ b n i.e. sin φ o φ sin b n Example 5 Find the angle between the line and the plane 0 x + y z 3. x + y z b n Solution Let θ be the angle between the line and the nomal to the plane. Conveting the given equations into vecto fom, we have and ( i + 3 k + λ ( i + 3 j + 6 k ( 0 i + j k 3 Hee b i + 3 j + 6 k and n 0 i + j k sin φ ( i + 3 j + 6 k (0 i + j k o φ 8 sin Fig.0 not to be epublished b n

31 THREE D IMENSIONAL G EOMETRY 493 EXERCIS E.3. In each of the following cases, detemine the diection cosines of the nomal to the plane and the distance fom the oigin. (a z (b x + y + z (c x + 3y z 5 (d 5y Find the vecto equation of a plane which is at a distance of 7 units fom the oigin and nomal to the vecto 3 i + 5 j 6 k. 3. Find the Catesian equation of the following planes: (a (c ( i + j k (b ( i + 3 j 4 k [( s t i + (3 t j + ( s + t k ] 5 4. In the following cases, find the coodinates of the foot of the pependicula dawn fom the oigin. (a x + 3y + 4z 0 (b 3y + 4z 6 0 (c x + y + z (d 5y Find the vecto and catesian equations of the planes (a that passes though the point (, 0, and the nomal to the plane is i + j k. (b that passes though the point (,4, 6 and the nomal vecto to the plane is + i j k. 6. Find the equations of the planes that passes though thee points. (a (,,, (6, 4, 5, ( 4,, 3 (b (,, 0, (,,, (,, 7. Find the intecepts cut off by the plane x + y z Find the equation of the plane with intecept 3 on the y-axis and paallel to ZOX plane. 9. Find the equatio n of t he plane tho ugh th e intesection of the planes 3x y + z 4 0 and x + y + z 0 and the point (,,. 0. Find the vecto equation of the plane passing though the intesection of the planes.( 3 i + j k 7,.( i + 5 j + 3 k 9 and though the point (,, 3.. Find the equation of the plane tho ugh the line of intese ction of the planes x + y + z and x + 3y + 4z 5 which is pependicula to the plane x y + z 0. not to be epublished

32 494 MATHE MATI CS. Find the angle between the planes whose vecto equations ae ( i + j 3 k 5 and (3 i 3 j + 5 k In the following cases, detemine whethe the given planes ae paallel o pependicula, and in case they ae neithe, find the angles between them. (a 7x + 5y + 6z and 3x y 0z (b x + y + 3z 0 and x y (c x y + 4z and 3x 3y + 6z 0 (d x y + 3z 0 and x y + 3z (e 4x + 8y + z 8 0 and y + z In the following cases, find the distance of each of the given points fom the coesponding given plane. Point Plane (a (0, 0, 0 3x 4y + z 3 (b (3,, x y + z (c (, 3, 5 x + y z 9 (d ( 6, 0, 0 x 3y + 6z 0 Miscellaneous Examples Example 6 A line makes angles α, β, γ and δ with the diagonals of a cube, pove that cos α + cos β + cos γ + cos δ 4 3 Solution A cube is a ectangula paallelopiped having equal length, beadth and height. Let OADBFEGC be the cube with each side of length a units. (Fig. The fou diagonals ae OE, AF, BG and CD. T he diection cosines of the diagonal OE which is the line joining two points O and E ae ( a, 0, a G a 0 a 0 a 0,, a a a a a a a a a i.e., ,, X Z C(0, 0, a F(0, a, a not to be epublished Y O B(0, a, 0 D( a, a, 0 Fig.

33 THREE D IMENSIONAL G EOMETRY 495 Similaly, the diection cosines of AF, BG and CD ae 3,, ;, , and,, , espectively. Let l, m, n be the diection cosines of the given line which makes angles α, β, γ, δ with OE, AF, BG, CD, espectively. Then cosα 3 (l + m+ n; cos β ( l + m + n; 3 cosγ 3 (l m + n; cos δ (l + m n (Why? 3 Squaing and adding, we get cos α + cos β + cos γ + cos δ 3 [ (l + m + n + ( l + m + n ] + (l m + n + (l + m n ] 3 [ 4 (l + m + n ] 3 4 (as l + m + n Example 7 Find the equation of the plane that contains the point (,, and is pependicula to each of the planes x + 3y z 5 and x + y 3z 8. Solution T he equation of the plane containing the given point is A (x + B(y + + C (z 0... ( Applying the condition of pependiculaly to the plane given in ( with the planes x + 3y z 5 and x + y 3z 8, we have A + 3B C 0 and A + B 3C 0 Solving these equations, we find A 5C and B 4C. Hence, the equied equation is 5C (x + 4 C (y + + C(z 0 i.e. 5x 4y z 7 not to be epublished Example 8 Find the distance between the point P(6, 5, 9 and the plane detemined by the points A (3,,, B (5,, 4 and C(,, 6. Solution Let A, B, C be the thee points in the plane. D is the foot of the pependicula dawn fom a point P to the plane. PD is the equied distance to be detemined, which uuu uuu uuu is the pojection of AP on AB AC.

34 496 MATHE MATI CS uuu uuu uuu Hence, PD the dot poduct of AP with the unit vecto along AB AC. uuu So AP 3 i + 6 j + 7 k i uuu uuu and AB j k AC 3 i 6 j + k uuu uuu 3 i 4 j + 3 k Unit vecto along AB AC 34 Hence PD ( 3 i + 6 j + 7 k. 3 i 4 j + 3 k Altenatively, find the equation of the plane passing though A, B and C and then compute the distance of the point P fom the plane. Example 9 Show that the lines and x a + d α δ y a z a d α α + δ x b + c y b z b c ae coplana. β γ β β + γ Solution Hee x a d x b c y a z a + d a α δ b α c α + δ Now conside the deteminant x x y y z z a b c a b c y b z b + c a β γ b β not to be epublished c β + γ 34 b c a + d b a b + c a d α δ α α + δ β γ β β + γ

35 Adding thid column to the fist column, we get b a b a b + c a d α α α + δ β β β + γ THREE D IMENSIONAL G EOMETRY 497 Since the fist and second columns ae identical. Hence, the given two lines ae coplana. Example 30 Find the coodinates of the point whee the line though the points A (3, 4, and B(5,, 6 cosses the XY-plane. Solution The vecto equation of the line though the points A and B is 3 i + 4 j + k + λ [ (5 3 i + ( 4 j + (6 k ] i.e. 3 i + 4 j + k + λ ( i 3 j + 5 k... ( Let P be the point whee the line AB cosses the XY-plane. Then the position vecto of the point P is of the fom x i + y j. T his point must satisfy the equation (. (Why? i.e. x i + y j (3 + λ i + ( 4 3 λ j + ( + 5 λ k Equating the like coefficients of i, j and k, we have x 3 + λ y 4 3 λ λ Solving the above equations, we get x 3 3 and y Hence, the coodinates of the equied point ae, 5 0 3, 0. 5 Miscellaneous Execise on Chapte not to be epublished. Show that the line joining the oigin to the point (,, is pependicula to the line detemined by the points (3, 5,, (4, 3,.. If l, m, n and l, m, n ae the diection cosines of two mutually pependicula lines, show that the diection cosines of the line pependicula to both of these m n m n, n l n l, l m l m ae

36 498 MATHE MATI CS 3. Find the angle between the lines whose diection atios ae a, b, c and b c, c a, a b. 4. Find the equation of a line paallel to x-axis and passing though the oigin. 5. If the coodinates of the points A, B, C, D be (,, 3, (4, 5, 7, ( 4, 3, 6 and (, 9, espectively, then find the angle between the lines AB and CD. x y z 3 x y z 6 6. If the lines and ae pependicula, 3 k 3k 5 find the value of k. 7. Find the vecto equation of the line passing though (,, 3 and pependicula to the plane. ( i + j 5 k Find the equation of the plane passing though (a, b, c and paallel to the plane + + ( i j k. 9. Find the shotest distance between lines 6 i + j + k + λ ( i j + k and 4 i k + µ (3 i j k. 0. Find the coodinates of the point whee the line though (5,, 6 and (3, 4, cosses the YZ-plane.. Find the coodinates of the point whee the line though (5,, 6 and (3, 4, cosses the ZX-plane.. Find the coodinates of the point whee the line though (3, 4, 5 and (, 3, cosses the plane x + y + z Find the equation of the plane passing though the point (, 3, and pependicula to each of the planes x + y + 3z 5 and 3x + 3y + z If the poin ts (,, p and ( 3, 0, be equidist ant fom the plane + + (3 i 4 j k 3 0, then find the value of p. 5. Find the equation of the plane passing though the line of intesection of the planes ( i + j + k and ( i + 3 j k and paallel to x-axis. 6. If O be the oigin and the coodinates of P be (,, 3, then find the equation of the plane passing though P and pependicula to OP. not to be epublished 7. Find the equation of the plane which contains the line of intesection of the planes ( i + j + 3 k 4 0, ( i + j k and which is pependicula to the plane (5 i + 3 j 6 k

37 THREE D IMENSIONAL G EOMETRY Find the distance of the point (, 5, 0 fom the point of intesection of the line i j + k + λ (3 i + 4 j + k and the plane ( i j + k Find the vecto equation of the line passing though (,, 3 and paallel to the planes ( i j + k 5 and (3 i + j + k Find the vecto equation of the line passing though the point (,, 4 and pependicula to the two lines: x 8 y + 9 z 0 x 5 y 9 z 5 and Pove that if a plane has the intecepts a, b, c and is at a distance of p units fom the oigin, then + +. a b c p Choose the coect answe in Execises and 3.. Distance between the two planes: x + 3y + 4z 4 and 4x + 6y + 8z is (A units (B 4 units (C 8 units (D 3. The planes: x y + 4z 5 and 5x.5y + 0z 6 ae (A Pependicula (C intesect y-axis Summay (B Paallel (D passes though 5 0,0, 4 9 units Diection cosines of a line ae the cosines of the angles made by the line with the positive diections of the coodinate axes. If l, m, n ae the diection cosines of a line, then l + m + n. Diection cosines of a line joining two points P(x, y, z and Q(x, y, z ae x x y y z z,, PQ PQ PQ not to be epublished whee PQ ( ( x + z z x + ( y y Diection atios of a line ae the numbes which ae popotional to the diection cosines of a line. If l, m, n ae the diection cosines and a, b, c ae the diection atios of a line

38 5 0 0 MATHE MATI CS then a b c l ; m ; n a + b + c a + b + c a + b + c Skew lines ae lines in space which ae neithe paallel no intesecting. They lie in diffeent planes. Angle between skew lines is the angle between two intesecting lines dawn fom any point (pefeably though the oigin paallel to each of the skew lines. If l, m, n and l, m, n ae the diection cosines of two lines; and θ is the acute angle between the two lines; then cosθ l l + m m + n n If a, b, c and a, b, c ae the diection atios of two lines and θ is the acute angle between the two lines; then cosθ a a + b b + c c a b c a b c Vecto equation of a line that passes though the given point whose position vecto is a and paallel to a given vecto b is a + λ b. Equation of a line though a point (x, y, z and having diection cosines l, m, n is x x y y z z l m n The vecto equation of a line which passes though two points whose position vectos ae a and b is a + λ ( b a. Catesian equation of a line that passes though two points (x, y, z and x x (x, y, z is y y z z. x x y y z z If θ is th e acute angle between a + λb an d a + λb, th en b b cosθ b b not to be epublished If x x y y z z x x y y z z and l m n l m n ae the equations of two lines, then the acute angle between the two lines is given by cos θ l l + m m + n n.

39 THREE D IMENSIONAL G EOMETRY 50 Shotest distance between two skew lines is the line segment pependicula to both the lines. Shotest distance between a +λ b and a +µ b is ( b b ( a a b b Shotest distance between the lines: x x y y z z and a b c x x y y z z a b c is x x y y z z a b c a b c + + ( b c b c ( c a c a ( a b a b +λ +µ Distance between paallel lines a b and a b is b ( a a b In the vecto fom, equation of a plane which is at a distance d fom the oigin, and n is the unit vecto nomal to the plane though the oigin is n d. Equation of a plane which is at a distance of d fom the oigin and the diection cosines of the nomal to the plane as l, m, n is lx + my + nz d. The equation of a plane though a point whose position vecto is a and pependicula to the vecto N u u is ( a. N 0. not to be epublished Equation of a plane pependicula to a given line with diection atios A, B, C and passing though a given point (x, y, z is A (x x + B (y y + C (z z 0 Equation of a plane passing though thee non collinea points (x, y, z,

40 5 0 MATHE MATI CS (x, y, z and (x 3, y 3, z 3 is x x x x 3 x x y y y 3 y y y z z z z 3 z z Vecto equation of a plane that contains thee non collinea points having position vectos a b, and c is ( a.[( b a ( c a ] 0 Equation of a plane that cuts the coodinates axes at (a, 0, 0, (0, b, 0 and (0, 0, c is x y + a b z + c Vecto equation of a plane t hat p asses tho ugh t he in tesection of planes n d and n d is ( n + λ n d + λd, whee λ is any nonzeo constant. Catesian equation of a plane that passes though the intesection of two given planes A x + B y + C z + D 0 and A x + B y + C z + D 0 is (A x + B y + C z + D + λ(a x + B y + C z + D 0. Two lines a + λb and a + µ b ae coplana if ( a a ( b b 0 In the catesian fom above lines passing though the points A (x, y, z and ( x, y, z B y y z z ae coplana if b C x x a a y 0 y b b z z In the vecto fom, if θ is the angle between the two planes, n d and n n d, then θ cos n. n n The angle φ between the line a + λ b and the plane n d is not to be epublished c c 0.

41 THREE D IMENSIONAL G EOMETRY 503 b n sin φ b n The angle θ between the planes A x + B y + C z + D 0 and A x + B y + C z + D 0 is given by A A + B B + C C cos θ A + B + C A + B + C The distance of a point whose position vecto is a fom the plane n d is d a n The distance fom a point (x, y, z to the plane Ax + By + Cz + D 0 is Ax + By + Cz + D. A + B + C NCERT not to be epublished