Equations of lines in

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1 Roberto s Notes on Linear Algebra Chapter 6: Lines, planes an other straight objects Section 1 Equations of lines in What ou nee to know alrea: The ot prouct. The corresponence between equations an graphs. General concepts relate to linear functions. What ou can learn here: How to escribe a line in the Cartesian plane b using linear algebra notation an operations. Both in high school an in the calculus course ou have worke extensivel with lines in the Cartesian plane. Tell me about it! So, ou have seen the connections that exists between the equation of a line an its graph, especiall its slope an its position in the plane. An the intercepts an the other stuff. It is now time to explore more eepl these connections between lines an the linear equations we have encountere so far. We shall o that b using several of the tools that we have evelope so far an b constructing several other was to escribe a line algebraicall. Hence linear algebra! Exactl! You ma remember that, at the most elementar level, a line is escribe as a straight curve, that is, a curve with constant slope. This propert allows us to construct several tpes of equations that ientif or escribe a line. Here is a reminer of the forms that are most commonl use an that we have most likel seen in the past. Knot on our finger The slope-intercept equation of a line: Is of the form mx b. Is calle this wa because it inclues the slope m an the -intercept b explicitl. Applies onl to non-vertical lines. Is also calle functional equation, since it is given in the form f x. Linear Algebra Chapter 6: Lines, planes an other straight objects Section 1: Equations of lines in Page 1

2 Knot on our finger The point-slope equation of a line: Is of the form mx x. Is calle this wa because it inclues the slope m an one point x, explicitl. Applies onl to non-vertical lines. Knot on our finger The general equation of a line: Is of the form ax b c or ax b c. Is calle this wa because it can be constructe from an suitable information. Applies to all lines. Knot on our finger The two-point equation of a line: Can be written in one of the two forms x x x x x x1 1 x x1 Is calle this wa because it inclues two ifferent points x, an x, explicitl. 1 1 Applies to all lines, at least in the first form. Example: x 3 This line has slope m an -intercept, 3, as clearl visible in the formula. B using some basic algebra we can change this form into the general equation: x 3 x 3 From either equation, we can see that the point, 1 is also on the line, so we can use this point an the slope we alrea know to construct the point-slope form: 1 x Again some simple algebra can change this back to the general form. Also, we can use the points, 1 an 13 x 3 1 This also turns into the general forms in two eas steps.,1 3, 3 to get the two-points equation: 3, 3 Linear Algebra Chapter 6: Lines, planes an other straight objects Section 1: Equations of lines in Page

3 This example illustrates what we can o for equations of lines that have a slope. But we have much fewer options for those lines that on t have a slope, namel vertical lines. Example: x 3 The vertical line through the point 3, oes not have a slope an hence we can onl use the general or two-point form. But how o we construct either? B observing that an point on the vertical line we are seeking must have the same x- coorinate, that is, it must have x 3. But this is the general equation, with a 1, b, c 3! Alternativel, the same observation implies that the point the same line an we can use the two-point form: x 3 6 x A little rearrangement will give us again the form x 3. Yeah, eah! Will we learn anthing new in this section? Linear Algebra Chapter 6: Lines, planes an other straight objects Section 1: Equations of lines in 3, is also on Of course, m impatient paawan! It turns out that a line can also be efine in terms of vectors, instea of points an slopes, thus giving rise to et more forms of the equation. An wh o we nee more forms: aren t the ones we know enough? The new forms of the equation of a line will open several new oors for new or more efficient uses. For instance: the will provie efficient an/or effective was of eveloping certain tpes of applications the can be use to exten the content of straight to higher imensions an, with that, of lots of nice properties of lines the will give us the possibilit to use matrices in relation to lines an other straight objects. Show me! Let s begin with a new efinition of a line that shies awa from the selfreferential concept of straight. This approach is relate to the earlier observation that using the slope is cumbersome when working with vectors, especiall in higher imensions. The line containing a P x, an point having irection vector consists of all 1 points X x, such Definition that the vector PX xp is parallel to. That is, this line consists of all points that satisf the equation: k x p or x x k k 1 Notice that, as usual, in this efinition the points X x, an P x, are ientifie with the vectors x x an x Example: x 3 4 k k This line contains the point instance, the point 6,1 is on it: just let k=3. P p respectivel. 3, 4 an has irection X 1. So, for Page 3

4 Is this equivalent to the usual efinition? Excellent question! Although the picture suggests clearl that it shoul be, in mathematics we better check these important etails. So, for our satisfaction, I will let ou fin that out in the Learning questions. What I want to show ou instea, is how we can use this efinition to construct several other formulae that can be useful an/or informative. Proof Technical fact The line containing P x, 1 an with irection can be written in the following forms: x x 1 k x p k x x k 1 k We onl nee to file with the efining formula an use some basic vector operation: x x k1 k x x k 1 x x 1 x x 1 k k This is the first form claime. If we go back to the names of the vectors involve, we obtain: x x 1 k k x p If we compute the linear combination on the right we get the last form: x x 1 x x k1 k k Just as the previous forms of the equation of a line have their own name, so o these ones. Definitions The equation: x p k x x 1 k is the vector equation of the line containing P x,. an with irection In this equation: x x is the variable vector x p is the reference vector k is the parameter is the irection vector. 1 1 Linear Algebra Chapter 6: Lines, planes an other straight objects Section 1: Equations of lines in Page 4

5 Interesting, but rather abstract. True, so here is a picture to make it more visual. Remember that we ientif the point P x, with the reference vector p x an similarl for the variable point an vector. What the vector equation tells us is that to reach the point X from the origin, we first reach the point P b using the vector p, then travel in the irection of with a suitable multiple of it. An before giving ou the customar examples, here is another efinition closel relate to the vector equation of a line. Definition If we write the vector equation of a line with each component as a separate equation, we obtain the parametric equations of the line: x x 1 x x k1 k k Notice that the parametric equations of a line ma be viewe as the solutions of a sstem with x an as leaing variables an k as a free variable. But more on this later. P p k x X Example: This line contains x 3 1 k 4 3, 4 an has irection 1. Therefore, above ou have its vector equation, while its parametric equations are: x3 k 4 k Example: x 3 This is the vertical line through the point 3, an parallel to. Therefore we can write its vector equation as: x 3 k Also, its parametric equation are: x 3 k Notice that the secon equation oes not reall give an information, since k can take an value an is not use for x. Therefore, the first equation is all we nee, but that is again x 3. To construct each of these equations ou picke one point an one vector, so what happens if we pick ifferent ones? Linear Algebra Chapter 6: Lines, planes an other straight objects Section 1: Equations of lines in Page 5

6 Technical fact Each line in has infinitel man versions of its vector an parametric equations. Example: x 3 1 k 4 This line contains 1. But it also contains, for instance, the point 4, 6 (let k=1). Moreover, the vector 36 has the same irection as 1. Therefore, this line can also be escribe b the following equations: 3, 4 an has irection x 4 3 x 4 3t t t Notice that I have use a ifferent letter for the parameter (t instea of k) to emphasize that we are ealing with ifferent-looking equations, but remember that the represent the same line. An of course we can pla the same game with other points on the line an parallel vectors. I can see wh this is a convenient wa to escribe a line, since it explicitl uses a point an a irection vector. Exactl! An if ou like this, ou will also like the next form, which is also convenient, but for ifferent reasons. The ke motivation for the next form is given b the following observation: Proof Technical fact An vector perpenicular to a vector v1 v parallel to the vector n v v. v 1 v is Therefore, an vector parallel to n v is perpenicular to an line of vector equation x p kv. We onl nee notice that: n v v v v v v v v v v An here comes the interesting part Technical fact A line containing P x, 1 an with irection can be escribe b an equation of the form: x n p n In terms of components, this can be written as: x x 1 1 Linear Algebra Chapter 6: Lines, planes an other straight objects Section 1: Equations of lines in Page 6

7 Proof We nee to check two things: Ever point on the line satisfies the given equation. Ever point that satisfies the given equation is on the line. For the first item, if X x, k for which x p k. If we ot prouct both sies b the perpenicular vector n we get: is a point on the line, then there is a value of k k x n p n p n n p n This shows that the given equation is satisfie. On the other han, if x x satisfies the given equation, then, b using the istributivit of the ot prouct we can see that: x n p n x p n This means that the vector x-p is perpenicular to the perpenicular vector an hence is parallel to. Therefore: x p k x p k so that the point X is on the line. This eserves a efinition. The equation x n Definition p n is calle the normal p x an equation of the line containing with irection. 1 Example: This line contains equation is: x 3 1 k 4 3, 4 an has irection x Therefore, its normal This looks like a shrinking illusion: the equations are getting smaller! That is part of the beaut of the normal equation: its being so short an its hiing so much information behin it. For instance, the normal equation hies a goo ol frien: Technical fact If x a b x a b is the normal equation of a line, then its general equation is: ax b ax b If ax b c is the general equation of a line, then its normal equation can be written as: x a b c / b a b or as: if b x a b c / a a b if a Linear Algebra Chapter 6: Lines, planes an other straight objects Section 1: Equations of lines in Page 7

8 Proof The first statement is clear once we compute the ot proucts involve. For the secon, if b, so that the line is not vertical, we can change the general form to the slope-intercept form: a c x b b a This means that its slope is an hence its irection vector can be b chosen to be b a. But this means that the normal vector is a b. c Moreover, the point, is on the line an the normal equation b follows. A similar argument proves the secon case. This leas to a useful connection. Knot on our finger The coefficients of the general equation of a line are the components of a vector normal to the line. 5 Example: 7 x 3 This line contains Therefore, its general form is: 3, 7, has slope m an slope-intercept form: x 7 x x 9 x 5 9 This means that one of its normal forms is x I see that the normal form is not unique either, eh? Of course, since there are lots of options available when picking the reference point an the normal vector. To finish, let s see a neat geometrical use of these equations. Do ou know how to compute the istance from a point to a line in Let s see, I can fin the minimum of the istance function b using calculus You certainl can an there are other convolute was of oing it, but here is a ver simple metho base on the linear algebra formulae we have seen so far. Proof Technical fact The istance from a point P x, to a line ax b c is given b: The istance from P x, ax b c a b? to the line is, b efinition, the length of the perpenicular segment from P to the line, as shown in the following picture. Linear Algebra Chapter 6: Lines, planes an other straight objects Section 1: Equations of lines in Page 8

9 P as claime. proj n v ax b ax b ax b c a b a b Example: 3x 5 But if we pick a point Q x, on the line an point an compute the scalar projection of the vector v from Q to P onto the vector normal to the line we get exactl such length. Here is the picture: n P v Q To fin the istance from the point (, 4) to this line we first rewrite it in general form as 3x 5 an then appl the formula: ax b c a b 3 1 Cool, but what oes this all have to o with sstems an matrices? Lots: the connections become more apparent in higher imensions, but the start here. Remember that a sstem is obtaine b consiering several linear equations at the same time. Well, what happens when we consier several lines at the same time? 1 We know that v x x an that n ab vn x x a b proj n v n a b ax ax b b a b Since Q is on the line we know that: ax b c c ax b an therefore:. Therefore: Knot on our finger Given n lines in of general equations: a x b c,1i n i i i their common points of intersection are exactl the solutions of the sstem whose augmente matrix is: a1 b1 c an bn c n Linear Algebra Chapter 6: Lines, planes an other straight objects Section 1: Equations of lines in Page 9

10 Well, of course! Mabe, but ou ma want to check that ou are clear on this b aressing the Learning question that asks ou to explain wh this is true. An here is the first important consequence: this corresponence provies a first connection between lines an sstems. Proof Technical fact The set consisting of the points of intersection of n lines in contains either: no point, or a single point, or infinitel man points This follows immeiatel from what we know about the number of solutions of a sstem. Here are some examples that highlight the corresponence between the number of solutions of the sstem an the geometrical relation between the lines. Example: 3x 4 an x 14 The lines have one point of intersection, as is confirme b checking the number of solution of the sstem 3x x R R R 3R Since the augmente matrix an the matrix of coefficients have the same rank an this equals the number of variables, there is a unique solution, as expecte. We can clearl see this in the graph as well Can ou fin the coorinates of this intersection? Example: 3x 5 5, 3x 5, 3x 5 1 The graph of these three lines suggests that the are parallel an their slopes confirm this. The matrix of coefficients has three equal rows, two of which can be eliminate when using Gauss-Joran, but the constants are not the same, thus creating a leaing coefficient in the last column: no solutions, as expecte. However, let us consier these three equations, which are ver similar to the previous ones: 3x 5 5, 6x 1 1, 5 3x 5 This time we notice that the equations are multiples of each other an hence ientif the same line. The augmente matrix of the corresponing sstem has an REF with a single non-zero row, hence one free variable an infinitel man solutions: all the points of the line! This is a egenerate case, but it will become more meaningful in higher imensions. But now, let us look at a ifferent situation, which contains a etail that tens ot trick some stuents into error. Linear Algebra Chapter 6: Lines, planes an other straight objects Section 1: Equations of lines in Page 1

11 Example: x8 3, x 14, 3x 5 These equations ientif the three lines shown here. The augmente matrix is as follows: R R R3 3R The last two rows are not multiples of each other an hence will generate a leaing coefficient in the last column: no solutions. But wait a minute: on t we see three points of intersection? CAREFUL! We are looking for points that are common to ALL lines, not just two of them. An there is no point that belongs to all three lines. That s enough material for one section: make sure to igest it properl before moving on to the next one an some interesting generalizations. Summar B using vector notation, we can evise aitional was to ientif a line through an equation. There are several such notations (especiall vector, normal an parametric) an each of them highlights a particular feature of the line. B using linear sstems, we can ientif the points of intersection that are common to a set of lines. Common errors to avoi Make sure to istinguish among the ifferent tpes of equations of lines an their visible features. Mixing them up can easil lea to errors of interpretation an computation. Linear Algebra Chapter 6: Lines, planes an other straight objects Section 1: Equations of lines in Page 11

12 Learning questions for Section LA 6-1 Review questions: 1. Describe the ifferent tpes of equations for a line in.. Ientif the features of a line that are explicitl present in each tpe of equation of a line. 3. For each tpe of equation, ientif one avantage an one isavantage. 4. Explain the relation between the components of the normal vector of a line an the coefficients of the general equation of that line. Memor questions: 1. What is the general form of the slope-intercept equation of a line in? 5. What is the general form of the vector equation of a line in?. What is the general form of the point-slope equation of a line in? 6. What is the general form of the parametric equation of a line in? 3. What is the general form of the two-point equation of a line in? 7. What is the general form of the normal equation of a line in? 4. What is the general form of the general equation of a line in? Computation questions: 1. For the line containing the point, 5 an having slope 4 3, construct: a) The point-slope equation b) The general equation c) The normal equation ) The vector equation e) The parametric equations. Given the line of equations x 1 4, etermine: a) The coorinates of a point p on the line b) The components of a irection vector c) The value of the coefficient k for which 8, 7 pk 3. Fin the intersection of the lines 3x 1 an x3 5 Linear Algebra Chapter 6: Lines, planes an other straight objects Section 1: Equations of lines in Page 1

13 4. For each of the following sstems, escribe the geometric interpretation of each equation an of the solution set of the sstem. a) b) c) 5x 8 3x6 5x 8 5x 5x 8 1 x Use a linear algebra metho to compute the istance from the point,1 to the line 3x. Theor questions: 1. When two equations represent two non-parallel lines in R, what oes the solution of the sstem consisting of those equations represent?. When two equations represent two parallel lines in R, what oes the solution of the sstem consisting of those equations represent? 3. When two equations representing two lines in R are multiples of each other, what o the solutions of the sstem consisting of those equations represent? 4. Wh are the slope-intercept an the point-slope equations not suitable for vertical lines? Proof questions: 1. Prove that the equations of a line provie in this section are consistent with the usual efinition of a line, in the sense that the slope between an two points that satisf an one of those equations is alwas the same. Linear Algebra Chapter 6: Lines, planes an other straight objects Section 1: Equations of lines in Page 13

14 Template questions: 1. Pick a point an a irection vector an construct the equation of the corresponing line in all forms seen in this section.. Construct the general equations of two lines an fin their point of intersection, if an. 3. Pick a point an a line an compute the istance between the two. 4. Construct the equation of a line in an of the forms seen in this section an from it construct the other forms. What questions o ou have for our instructor? Linear Algebra Chapter 6: Lines, planes an other straight objects Section 1: Equations of lines in Page 14