EQUATIONS OF LINES AND PLANES


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1 EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint form, symmetric equtions of line, degenerte cses where direction vector hs one or more coordinte zero, intersecting lines, eqution of plne, ngle etween plnes, line of intersection of plnes, distnce of point nd plne. Wht students should hopefully get: How the eqution setup reltes to the generl setup for curves nd surfces. Understnding of the degenerte cses. Role of prmeter restrictions in defining line segment. Deeper understnding of reltionship of direction vector nd direction cosines Direction cosines. Executive summry (1) For nonzero vector v, there re two unit vectors prllel to v, nmely v/ v nd v/ v. (2) The direction cosines of v re the coordintes of v/ v. if v/ v = l, m, n, then the direction cosines re l, m, nd n. We hve the reltion l 2 + m 2 + n 2 = 1. Further, if α, β, nd γ re the ngles mde y v with the positive x, y, nd z xes, then l = cos α, m = cos β, nd n = cos γ Lines. Words... (1) A line in R 3 hs dimension 1 nd codimension 2. A prmetric description of line thus requires 1 prmeter. A topdown equtionl description requires two equtions. (2) Given point with rdil vector r 0 nd direction vector v long line, the prmetric description of the line is given y r(t) = r 0 + tv. If r 0 = x 0, y 0, z 0 nd v =,, c, this is more explicitly descried s x = x 0 + t, y = y 0 + t, z = z 0 + tc. (3) Given two points with rdil vectors r 0 nd r 1, we otin vector eqution for the line s r(t) = tr 1 + (1 t)r 0. If we restrict t to the intervl [0, 1], then we get the line segment joining the points with these rdil vectors. (4) If the line is not prllel to ny of the coordinte plnes, this prmetric description cn e converted to symmetric equtions y eliminting the prmeter t. With the ove nottion, we get: = z z 0 c This is ctully two equtions rolled into one. (5) If c = 0 nd 0, the line is prllel to the xyplne, nd we get the equtions:, z = z 0 Similrly for the other cses where precisely one coordinte is zero. (6) If = = 0 nd c 0, the line is prllel to the zxis, nd we get the equtions: Actions... x = x 0, y = y 0 (1) To intersect two lines oth given prmetriclly: Choose different letters for prmeters, equte coordintes, solve 3 equtions in 2 vriles. Note: Expected dimension of solution spce is 2 3 = 1. 1
2 (2) To intersect line given prmetriclly nd line given y equtions: Plug in the coordintes s functions of prmeters into oth equtions, solve. Solve 2 equtions in 1 vrile. Note: Expected dimension of solution spce is 1 2 = 1. (3) To intersect two lines given y equtions: Comine equtions, solve 4 equtions in 3 vriles. Note: Expected dimension of solution spce is 3 4 = Plnes. Words... (1) Vector eqution of plne (for the rdil vector) is n (r r 0 ) = 0 where n is norml vector to the plne nd r 0 is the rdil vector of ny fixed point in the plne. This cn e rewritten s n r = n r 0. Using n =,, c, r = x, y, z, nd r 0 = x 0, y 0, z 0, we get the corresponding sclr eqution x + y + cz = x 0 + y 0 + cz 0. Set d = (x 0 + y 0 + cz 0 ) nd we get x + y + cz + d = 0. (2) The direction or prllel fmily of plne is determined y its norml vector. The ngle etween plnes is the ngle etween their norml vectors. Two plnes re prllel if their norml vecors re prllel. And so on. Actions... (1) Given three noncolliner points, we find the eqution of the unique plne contining them s follows: first we find norml vector y tking the cross product of two of the difference vectors. Then we use ny of the three points to clculte the dot product with the norml vector in the ove vector eqution or the corresponding sclr eqution. Note tht if the points re colliner, there is no unique plne through them ny plne contining their line is plne contining them. (2) We cn compute the ngle of intersection of two plnes y computing the ngle of intersection of their norml vectors. (3) The line of intersection of two plnes tht re not prllel cn e computed y simply tking the equtions for oth plnes. This, however, is not stndrd form for line in R 3. To find stndrd form, either find two points y inspection nd join them, or find one point y inspection nd nother point y tking the cross product of the norml vectors to the plne. (4) To intersect plne nd line, plug in prmetric expressions for the coordintes rising from the line into the eqution of the plne. We get one eqution in the one prmeter vrile. In generl, this is expected to hve unique solution for the prmeter. Plug in the vlue of the prmeter into the prmetric expressions for the line nd get the coordintes of the point of intersection. (5) For point with coordintes (x 1, y 1, z 1 ) nd plne x + y + cz + d = 0, the distnce of the point from the plne is given y x 1 + y 1 + cz 1 + d / c Lines nd plnes 1.1. Lines: dimension nd codimension. A line in R n hs dimension one nd codimension n 1. In prticulr, line in Eucliden spce R 3 hs dimension 1 nd codimension 3 1 = 2. In prticulr, sed on wht we know of dimension nd codimension, we expect tht: In topdown or reltionl description, we should need two independent equtions to define line. In ottomup or prmetric description, we should need one prmeter to define line Plnes: dimension nd codimension. A plne in R 3 is 2dimensionl, nd it hs codimension 3 2 = 1. In prticulr, sed on wht we know of dimension nd codimension, we expect tht: In topdown or reltionl description, we should need one eqution to define plne. In ottomup or prmetric description, we should need two prmeters to define plne. This gets into the relm of functions of two vriles, so we will defer the ctul 2prmeter description of plnes for now Intersection theory. We hve the following sic intersection fcts: Intersect Generic cse Specil cse 1 Specil cse 2 Specil cse 3 Plne, plne Line Empty (prllel plnes) Plne (equl plnes) Plne, line Point Empty (line prllel to, not on plne) Line (line on plne) Line, line Empty (skew lines) Point (intersecting lines) Empty (prllel lines) Line (equl lines) 2
3 The generic cse here represents the cse tht is most likely, i.e., the cse tht would rise if the things eing intersected were chosen rndomly. There re mthemticl wys of mking this precise, ut these re eyond the current scope. In prticulr, it is worth pointing out tht the generic cse is exctly s intersection theory predicts. Let s consider the three generic cses: Generic intersection of plne nd plne: A plne hs codimension 1, so the intersection of two plnes (genericlly) hs codimension 1+1 = 2. We know tht line hs codimension 2, so this mkes sense. Generic intersection of plne nd line: A plne hs codimension 1 nd line hs codimension 2, so the intersection of plne nd line (genericlly) hs codimension 1+2 = 3, so it is zerodimensionl. A point is zerodimensionl. Generic intersection of line nd line: A line hs codimension 2, so the intersection of two lines (genericlly) hs codimension = 4, so it hs dimension 3 4 = 1. Negtive dimension indictes tht the intersection is genericlly empty. After we study the intersection theory in detil for lines nd plnes, we will e in position to cquire etter understnding of the generl principles of intersection theory. Specificlly, we will cquire etter grsp of the nongeneric cses where the intersections don t work out s they genericlly do. 2. Equtions of lines 2.1. The pointdirection form. The generl principle ehind this is the sme s it is with the pointslope form. Bsiclly, to descrie line, it suffices to specify point on the line, nd the direction of the line. The direction is specified y specifying ny vector prllel to the line. Specificlly, given line with points A nd B on it, the direction of the line is given y tking the vector AB. Note tht ny two vectors tht re sclr multiples of ech other (i.e., prllel to ech other) specify the sme direction. Suppose r 0 is the rdil vector for one point on the line, nd v is ny nonzero vector long the line. Then the rdil vector (i.e., vector from the origin to point) for points on the line cn e defined y the prmetric eqution: r(t) = r 0 + tv where t vries over the rel numers. For ech vlue of t, we get rdil vector for some point on the line, nd every point on the line is covered this wy. Suppose r 0 = x 0, y 0, z 0 nd v =,, c. Then r 0 + tv is the vector: x 0 + t, y 0 + t, z 0 + tc The corresponding prmetric description of curve is: {(x 0 + t, y 0 + t, z 0 + tc) : t R} Note tht the choice of prmetric description depends on the choice of sepoint in the line nd the choice of vector (which cn e vried up to sclr multiples). By the wy, here is some terminology (which we overlooked erlier). The direction cosines for prticulr direction re defined s the coordintes of the unit vector in tht direction. The direction cosines of prticulr direction re denoted l, m, nd n. For instnce, if direction vector is 1, 2, 3, then the corresponding unit vector is 1/ 14, 2/ 14, 3/ 14, so the direction cosines re l = 1/ 14, m = 2/ 14, nd n = 3/ 14. The direction cosines re lso the cosines of the ngles mde y the vectors with the xxis, yxis, nd zxis. They stisfy the reltion: l 2 + m 2 + n 2 = The twopoint form. Suppose r 0 nd r 1 re the rdil vectors of two points on line. Then, we cn get line in the pointdirection form y setting v = r 1 r 0. We thus get the form: This simplifies to: r(t) = r 0 + t(r 1 r 0 ) 3
4 r(t) = tr 1 + (1 t)r 0 As t vries over ll of R, this gives the whole line. When t = 0, we get the point with rdil vector r 0 nd when t = 1, we get the point with rdil vector r 1. If we llow only 0 t 1, we get the line segment joining the two points Topdown description: symmmetric equtions. To otin the symmetric equtions, we strt with the prmetric equtions nd then eliminte the prmeter. In other words, with the prmetric description: We note tht: {(x 0 + t, y 0 + t, z 0 + tc) : t R} x = x 0 + t, = t = Similrly, we get t = (y y 0 )/ nd t = (z z 0 )/c. Eliminting t, we get: = z z 0 c Note tht while this looks like single long eqution, it is ctully two equtions: nd y y 0 = z z 0 c This is in keeping with wht we expect/hope tht to descrie 1dimensionl suset in 3dimensionl spce, we need 3 1 = 2 equtions. Intuitively, wht these equtions re sying is tht the coordinte chnges re in the rtio : : c Exceptionl cse of lines prllel to one of the coordinte plnes. The symmetric equtions formultion reks down if one of the coordintes of the direction vector,, c is zero. In this cse, the line is prllel to one of the three coordinte plnes, with the third coordinte eing unchnged (e.g., if c = 0, then the line is prllel to the xyplne, ecuse its zcoordinte is unchnged). They rek down even more when two coordintes of the direction vector re zero, which mens tht the line is prllel to one of the xes. In this cse, the symmetric equtions given ove do not work, nd we insted do the following. If only one coordinte of the direction vector is zero: If c = 0 nd, 0, then we get the two equtions:, z = z 0 Similrly for the other cses. If two coordintes re zero: If, sy = = 0, then we get the two equtions: x = x 0, y = y 0 z does not pper in the equtions ecuse it cn vry freely. This line is prllel to the zxis. 4
5 2.5. Pirs of lines: questions out intersection. As we noted erlier, lines in R 3 hve codimension 2, so the intersection of two lines is expected to e empty. There re qulittively four possiilities: (1) The lines re skew lines: This is the most independent cse possile. Here, the equtions descriing the two lines re s independent of ech other s possile nd the two lines thus do not lie in the sme plne. They do not intersect. (2) The lines re intersecting lines in the sme plne: This is somewht less independent cse. Here, there is plne (not necessrily contining the origin) tht contins oth lines, nd the lines re not prllel, so they intersect t point. (3) The lines re prllel lines in the sme plne: Here, the equtions for the line re inconsistent in specific wy, so they lie in the sme plne ut re prllel. They do not intersect. Although the conclusion out intersection is the sme oth for pirs of prllel lines nd for pirs of skew lines, the resons ehind this conclusion re different. (4) The two lines re ctully the sme line: In this cse, their intersection is the sme line. This is the most dependent cse possile. We now exmine how to find the intersection of two lines. The pproch is simply specil cse of finding the intersection of two curves. Since the equtions re ll liner, we cn ctully devise specific procedures to solve the equtions. Both lines re given prmetriclly: In this cse, we first mke sure we hve different letters for the prmeters for ech line. Then we equte coordintewise nd solve the system of 3 liner equtions in 2 vriles (the prmeter vriles for the two lines). Note tht the numer of equtions is more thn the numer of vriles unsurprising since the generic cse is one of skew lines. After finding solutions for the two prmeters, plug ck to find the points. One line is given prmetriclly in terms of t, the other using symmetric equtions: We sustitute the prmetric expressions into the vlues of x, y, nd z in the symmetric equtions nd solve the system of two equtions in the one (prmeter) vrile t. After finding solutions for t, plug ck to find the points. Both lines re given y symmetric equtions: We solve ll the four symmetric equtions. 3. Plnes 3.1. Vector description in terms of dot product. For given plne in R 3, it either lredy psses through the origin, or there is unique plne prllel to it tht psses through the origin. We sy tht two plnes re prllel if they either coincide or they do not intersect equivlently, if for every line in one plne, there is line in the other plne prllel to it. A fmily of prllel plnes cn e thought of s shring direction. But how do we specify the direction of plne, which is twodimensionl oject? The ide is to look t the complement, or the codimension, of the plne. Specificlly, we look t the direction tht is orthogonl to the plne. There is unique direction vector (up to sclr multiples) orthogonl to fmily of prllel plnes. Further, the dot product of this vector with the rdil vector in ny fixed plne in the fmily is constnt, nd this constnt differs for ech plne in the fmily. This llows us to give equtions for plnes s follows. Let n e norml vector (orthogonl vector) to plne nd let r 0 e the rdil vector for fixed point in the plne. Then, if r is the rdil vector for n ritrry point in the plne, we hve: Rerrnging, we get: n (r r 0 ) = 0 n r = n r 0 Note tht the right side is n ctul rel numer. If n =,, c nd r 0 = x 0, y 0, z 0, we get the sclr eqution: x + y + cz = x 0 + y 0 + cz 0 If we define d = (x 0 + y 0 + cz 0 ), we cn rewrite the ove s: 5
6 x + y + cz + d = 0 Conversely, ny eqution of the ove sort, where t lest one of the numers,, nd c is nonzero, gives plne Plne prllel to the coordinte xes nd plnes. We sy tht plne nd line re prllel if either the line lies on the plne or they do not intersect t ll. If = 0, the plne is prllel to the xxis. If = 0, the plne is prllel to the yxis. If c = 0, the plne is prllel to the zxis. If = = 0, the plne is prllel to the xyplne. If = c = 0, the plne is prllel to the yzplne. If = c = 0, the plne is prllel to the xzplne Finding the eqution of plne given three points. To specify plne, we need to provide t lest three points on the plne. Given these three points, we cn find the eqution of the plne s follows: We first tke two difference vectors nd tke their cross product to find norml vector to the plne: If the points re P, Q, nd R, we tke the difference vectors P Q nd P R nd compute their cross product. We now use the vector eqution, nd hence from tht the sclr eqution, tking ny of of the three points P, Q, or R s the sepoint. Note tht if the three points given re colliner, then they do not define unique plne. Rther, ny plne through the line joining these three points works. It is no surprise tht the ove procedure fils t the stge where we need to tke cross product, ecuse the cross product turns out to e the zero vector Intersecting two plnes: line of intersection. Given two plnes, the typicl cse is tht they intersect in line. If we hve sclr equtions for oth plnes, then the intersection line cn e descried y tking the two equtions together. Unfortuntely, this pir of two equtions together, while it does define line, is not directly one of the stndrd descriptions of line. There re mny wys of otining the line in stndrd form. One of these is s follows: first, find norml vectors to the plnes. For instnce, if the equtions for the plnes re: 1 x + 1 y + c 1 z + d 1 = 0 2 x + 2 y + c 2 z + d 2 = 0 Then the norml vectors to these plnes re 1, 1, c 1 nd 2, 2, c 2. A direction vector long the line of intersection must e perpendiculr to oth these norml vectors, hence, it must e in the line of the cross product. Hence, we tke the cross product 1, 1, c 1 2, 2, c 2. Now tht we ve found the direction vector long the intersection of these plnes, we need to find just one point long the intersection nd we cn then use the pointdirection form. One wy of finding point is to set z = 0 in oth equtions nd solve the system for x nd y (this is ssuming tht neither is prllel to the xyplne; otherwise choose some other coordinte). Note tht if the plnes re prllel or coincide, then their norml vectors re prllel nd thus the cross product of the norml vectors ecomes zero. Conversely, the cross product ecoming zero mens the plnes re prllel, so there is good reson for the line of intersection to not mke sense Intersecting two plnes: ngle of intersection. The ngle of intersection etween two plnes is the ngle of intersection etween their norml vectors. As for the line of intersection, we cn extrct the norml vector from the sclr eqution of the plnes. To compute the ngle of intersection, we use the formul s rc cosine of the quotient of the dot product y the product of the lengths. 6
7 3.6. Intersecting plne nd line. Given plne nd line, we cn intersect them s follows: If the plne is given y sclr eqution nd the line is given prmetriclly using prmeter t, then to compute the intersection, we plug in ll coordintes s functions of the prmeter into the sclr eqution for the plne, nd solve one eqution in the one vrile t. After finding the solution t, we plug this into the prmetric eqution of the line to find the coordintes of the point of intersection. There re three possiilities: The typicl cse is tht we hve liner eqution in one vrile, nd it hs unique solution. In other words, the plne nd line intersect t point. Another cse is tht the eqution simplifies to something nonsensicl, such s 0 = 1. In this cse, there is no intersection. Geometriclly, this mens the line is prllel to ut not on the plne. The finl cse is tht the eqution simplifies to tutology, such s 0 = 0. In this cse, ll rel t give solutions. Geometriclly, this mens tht the line is on the plne Distnce of point from plne. We will not hve much occsion to use this formul, ut we note it riefly nonetheless. Given point with coordintes (x 1, y 1, z 1 ) nd plne x + y + cz + d = 0, the distnce from the point to the plne is given y the formul: x 1 + y 1 + cz 1 + d c 2 7
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