parameterization of a line through two points calculator

Further if two of the components of a direction vector are equal to 0, then there are no symmetric equations for the line. We will use the first so that \begin{align*} ♪ We can write our line parameterization as \begin{align*} (x,y,z) = (t+2, -2t-3, t). The graph of a curve in space. This is where you are in error. More in-depth information read at these rules. Just as a vector field is defined by a function that returns a vector, a scalar field is a function that returns a scalar, such as z = f . The Useful Way This way of parameterizing is useful because it allows us to choose our starting point, which direction we travel on the line, and how fast we go. Write down all three forms of the equation of the line. The parameterization x = h (t) x = h (t), y =g (t) y = g (t) will then determine an orientation for the curve where the positive direction is the direction that is traced out as t t increases. This load model divides the bridge's deck into a 3.0 m width centered virtual lane, where a distributed load of 9.0 kN/m 2 is applied, and two remaining . Parametrization of a Line Through Two Points 1,443 views Jun 12, 2017 5 Dislike Share Save Respite Math 57 subscribers Subscribe Using a similar idea of finding the slope-intercept form of a line. How to calculate the angles and sides of a triangle? Apply the formula for surface area to a volume generated by a parametric curve. Then press the 8 key and then ENTER, and the line . Recall that the slope of the line that makes angle with the positive -axis is given by t a n . Schwartz et al. Spherical to Cylindrical coordinates. Entering data into the equation of a line calculator. . Keep changing the parameter, and we . Plane and Parametric Equations in R 3 Calculator: Given a vector A and a point (x,y,z), this will calculate the following items: 1) Plane Equation passing through (x,y,z) perpendicular to A 2) Parametric Equations of the Line L passing through the point (x,y,z) parallel to A Simply enter vectors by hitting return after each vector entry (see vector page for an example) Homework Statement Find a parametrization of the vertical line passing through the point (7,-4,2) and use z=t as a parameter. Theorem 17. 1 Parametrized curve Parametrized curve Parametrized curve A parametrized Curve is a path in the xy-plane traced out by the point (x (t),y (t)) as the parameter t ranges over an interval I. Example 1 Write down the equation of the line that passes through the points \(\left( {2, - 1,3} \right)\) and \(\left( {1,4, - 3} \right)\). Find the parametric equations of the line throughthe two points (2,1,7)and (1,3,5). Since these two points are on the line the vector between them will . (b) Find a parameterization of C and a formula for F. Check your answer with the instructor. Find the point-slope equation of the line y = mx + b. Parametrize by letting x = t and y = mt + b. We begin with our familiar formula for arc length. Further if two of the components of a direction vector are equal to 0, then there are no symmetric equations for the line. You can use this calculator to solve the problems where you need to find the line equation that passes through the two points with given coordinates. Step 1: Find a set of equations for the given function of any geometric shape. Cartesian to Spherical coordinates. In order to understand how to parameterize line segments, students should understand the concept of parametric equations. radiative energy losses, it is possible to calculate the underground muon intensity Crouch curve. Since that time they have been applied to a wide range of problems, from the inversion of free oscillation. So in general we can say that a circle centered at the origin, with radius r, is the locus of all points that satisfy the equations. the cross product. The vector equation of the line segment is given by. To find a line perpendicular to a given line through a given point, use the point-slope form and plus in the perpendicular slope . A) a point that it actually passes through, say #vec a#, AND. . There are different forms of the equation of a circle: general form; standard form; parametric form; polar form. Plane equation given three points. Cylindrical to Cartesian coordinates Given the following fields F and curves C, write down which . Notice that, in the first set of equations, t= -2 gives (7, 4, 0) while, in the second set, t= 2 gives (7, 4, 2). b. The vector between any two points on the two lines is then. Learn more about distance; line to line The straight line distance from Knoxville, . x = r cos (t) y = r sin (t) A parameterization of a line has the form r ( t) = P + t D where P is a vector "touching" the line and D is a direction vector for the line. The parametric equations of a line are given by. B) a vector describing the direction in which it travels, say #vec b#..such that the line itself is . Cartesian to Cylindrical coordinates. Since the altitude A E passes through the point A ( - 3, 2), using the point-slope form of the equation of a line, the . . t. These integrals are known as line integrals over vector fields. There are lots of possible such vectors u and v. To find one such vector v, find the difference between any two points on the line. Step 1 However, the approaches taken there seem rather cumbersome to use for oriented curves that are not line segments. Since a line is determined by two points, the lines defined by those sets of equations are the same line. Shortest distance between a point and a plane. The collection of all points for the possible values of t yields a parametric curve that can be graphed. The line segments between (x0,y0) and (x1,y1) can be expressed as: x(t) = (1 −t)x0 +tx1. Line integrals are useful in physics for computing the work done by a force on a moving object. We've been using the two dimensional version of this over the last couple of sections. Additional features of equation of a line calculator. the ones that you go . The second calculator finds the normal vector perpendicular to two vectors, i.e. That is, if the point satisfies the equation of the circle, it lies on the circle's circumference. If you parameterize the curve such that you move in the opposite direction as increases, the value of the line integral is multiplied by . r ( t) = ( 1 − t) r 0 + t r 1 r (t)= (1-t)r_0+tr_1 r ( t) = ( 1 − t) r 0 + t r 1 . Parameterization and Modelling of Large Off-Road Tyres for Ride Analyses Part 2 - Parameterization and Validation of Tyre Models M. Find the area under a parametric curve. The parametric equation of a circle. this depends on your parameterization of the line. The relationship between the vector and parametric equations of a line segment. #vec r = vec a + lambda vec b qquad square# For #vec a#, we can simply choose the completely arbitrary point, so here we choose the point at which z = 0 Theory. The above two relations can also be written in a fancy way: (x - x1) / cosθ = (y - y1) / sinθ = r. And this is the parametric form of the equation of a straight line. From the above we can find the coordinates of any point on the circle if we know the radius and the subtended angle. So essentially, I want the equation-- if you're thinking in Algebra 1 terms-- I want the equation for the line that goes through these two points. The vector equation of a line is given by. This calculator calculates for the length of one side of a right triangle given the length of the other two sides. a 1 x+b 1 y+c 1 = 0. The direction vector of the plane orthogonal to the given lines is collinear or coincides with their direction vectors that is. d. ⁡. y(t) = (1 −t)y0 +ty1, where 0 ≤ t ≤ 1. Monte Carlo inversion techniques were first used by Earth scientists more than 30 years ago. Examples Example 1 a. The calculator will calculate the slope by finding the ratio of the "vertical change" (dy) to the "horizontal change" (dx) between two distinct points A, B on a line. Question: 14 of 19 Which of the following is a parameterization of the line that passes through the point (1.-3) with a slope of -27 H-2t and y-2t - 1. for any t want and y. Three forms of the equation of the line: d = √ (x2 - x1)2 + (y2 - y1)2. where (x 1, y 1) and (x 2, y 2) are the coordinates of the two points involved. By contrast, the line integrals we dealt with in Section 15.1 are sometimes referred to as line integrals over scalar fields. Since the x-coordinate of the given point is 8, the equation of the vertical line passing through the point is: x = 8. You can also use the calculators on the left. }\) Steps to Use Parametric Equations Calculator. Use and keys on keyboard to move between field in calculator. Notice that this parameterization involves two parameters, u and v, because a surface is two-dimensional, and therefore two variables are needed to trace out the surface.The parameters u and v vary over a region called the parameter domain, or parameter space —the set of points in the uv-plane that can be substituted into r.Each choice of u and v in the parameter domain gives a point on the . Click for vector b the vector product of v1 and v2; 5. Use and keys on keyboard to move between field in calculator. The coordinate form is an equation that gives connections between all the coordinates of . The parameterization should be at (7, 9) when t = 0 and should draw the line from right to left. Assume t = 0 corresponds to the given point, t increases . Solution —2) with . The point-normal form consists of a point and a normal vector standing perpendicular to the plane. Keywords: Math, shortest distance between two lines. Find two other points on the line. Spherical to Cylindrical coordinates. Theory. Finally, the load model for roadway traffic given by the European code EN 1991-2 (i.e., the LM71 model) was applied to calculate the structure's mechanical performance in its initial state. Question: 2. An unlimited number of examples may be generated interactively along with their solutions. If you insist on taking notes pause . Find a parameterization c(t) for the curve y=x^2 where c(0)=(3,9) I recognize this as a parabola going from left to right on a graph, but I am stuck in that I don't know how to incorperate c(0)=(3,9). Now what if I asked you, give me a parametrization of the line that goes through these two points. Solve advanced problems in Physics, Mathematics and Engineering. It is divided into Entering data into the equation of a line calculator. Shortest distance between two lines. be the vector that starts at the second point and ends at the first point. Step 1: Write an equation for a line through (7,5) with a slope of 3. That is, if you don't have a calculator handy, a 43% jump in our nation's money supply in just over two years. . I calculated the parametric function of L1 to (x,y)=(2t,3t), but I'm completely lost at how to calculate the same type of function . We'll first need the parameterization of the line segment. Below is shown a line through two points \( P(x_p,y_p,z_p) \) and \( Q(x_q,y_q,z_q) \). Step 3: Find out the value of a second variable . Enter coordinates of the first and second points, and the calculator shows both parametric and symmetric line equations. So the classic way, you would have figured out the slope and all of that, and then you would have . General Form Equation of . one of the original equations. [] and Timsari and Leahy [] compute quasi-isometric flat maps of the cerebral cortexDrury et al. They are review videos for my students. More in-depth information read at these rules. Articles that describe this calculator Equation of a line given two points Parametric line equation from two points First Point x y Second point x y Equation for x Equation for y Direction vector Calculation precision Digits after the decimal point: 2 Given two points on the line, P and Q, the equation r ( t) = P + t Q is not the correct parameterization. Brain surface parameterization has been studied intensively. A calculator and solver to find the equation of a line in vector, parametric and symmetric forms is presented. Instead of defining y in terms of x, parametric equations define both x and y in terms of a parameter t. Each value of t yields a point (x (t),y (t)) that can be plotted. The parametric equation consists of one point (written as a vector) and two directions of the plane. b. Parametrize the line that goes through the points (2, 3) and (7, 9). Find the vector, parametric, and symmetric equations of the line through PO (3, 7, vector (1, —3, 2). . The question is: Let L1 be the line passing through the origin and the point (2,3), and L2 be the line passing through the point (-1,0) and orthogonal (perpendicular) to L1. Three forms of the equation of the line: 2, 3π 2], (x(t) = 2acostcost = a(1+cos2t), y(t) = 2acostsint = asin2t. → v = (x1,y1) −(x0,y0) = (x1 −x0,y1 −y0). You can input only integer numbers or fractions in this online calculator. Math Expression Renderer, Plots, Unit Converter, Equation Solver, Complex Numbers, Calculation History. Putting values in the formula: 20 2 + 21 2 = c 2. The parametric equations of a line passing through two points. View Notes - Math 291 from MATH 291 at Rutgers University. Step 2: Then, Assign any one variable equal to t, which is a parameter. There are two steps to finding the fast parameterization. . value of t which minimizes the distance from the point P to line L; that is, calculate the value of t which minimizes f(t) from part (b). Distance from a point to a line is equal to length of the perpendicular distance from the point to the line. Previous Work. -2tan -1 for --> <t; + 3 and 4+1 for any 2t and HH4-2 for any This problem has been solved! * Write the vector, parametric and symmetrical of a line through a given point in a given direction, and a line through two data points. m= dxdy. At this point our only option for sketching a parametric curve is to pick values of t t, plug them into the parametric equations and then plot the points. (d)Compute the distance from the point P(5;3;0) to line L by . MATH 291 PROBLEM SOLVING This is a comprehensive guide on how to answer every type of question you'll face in MATH 291. A calculator and solver to find the equation of a line in vector, parametric and symmetric forms is presented. Volume of a tetrahedron and a parallelepiped. Method 1. Parallel lines are a fixed distance apart and will never meet, no matter how long they are extended. Plane equation given three points. = ΔxΔy. Additional features of equation of a line calculator. An equation of a circle is an algebraic way to define all points that lie on the circumference of the circle. Thus, we can think of the curve as a collection of terminal points of vectors emanating from the origin. See Parametric equation of a circle as an introduction to this topic. for the line, we will need . The origin of the polar system includes all vertical lines. Plane and Parametric Equations in R 3 Calculator: Given a vector A and a point (x,y,z), this will calculate the following items: 1) Plane Equation passing through (x,y,z) perpendicular to A 2) Parametric Equations of the Line L passing through the point (x,y,z) parallel to A Simply enter vectors by hitting return after each vector entry (see vector page for an example) Therefore, the point where they meet is called the origin. Following are four different fields with a curve passing through each field. Section 5-2 : Line Integrals - Part I. Free Online Scientific Notation Calculator. Show Next Step Example 5 Parametrize the line that goes through the points (2, 3) and (7, 9) so that it takes 3 steps to travel from one point to the other. In order to obtain the parametric equations of a straight line, we need to obtain the direction vector of the line. Cartesian to Spherical coordinates. The parametrization of a line is r(t) = u + tv, where u is a point on the line and v is a vector parallel to the line. Use the equation for arc length of a parametric curve. In this section we are now going to introduce a new kind of integral. Now that we have introduced the concept of a parameterized curve, our next step is to learn how to work with this concept in the context of calculus. Determine the vector parameterization of a line that passes through the point (3, 2, -1) when /- 2 seconds and the point (4, -3, 0) when 7 seconds. Sometimes we need to find the equation of a line segment when we only have the endpoints of the line segment. Spherical to Cartesian coordinates. It's a chili dog. The first calculator will directly compute the equation of a plane if you input the three points. Figure 9.6.1. Choose two points that are on the line. c. Is (—1, 19, 8) on the line? The direction vector from (x0,y0) to (x1,y1) is. ∘. This online calculator finds parametric equations for a line passing through the given points. 4.The line which passes through the point ( 2;1;4) and is parallel to both the xy-plane . What has changed 6. Spherical to Cartesian coordinates. However, before we do that it is important to note that you will need to remember how to parameterize equations, or put another way, you will need to be able to write down a set of parametric equations for a given curve. = x1. You know four different techniques for calculating line integrals: parameterization, along with the three named theorems: The Fundamental Theorem of Calculus, Green's Theorem, and Stoke's Theorem. You can input only integer numbers or fractions in this online calculator. Below is shown a line through two points \( P(x_p,y_p,z_p) \) and \( Q(x_q,y_q,z_q) \). In order to graph this line on the calculator, press 2nd, then PRGM [DRAW], and select 4:Vertical. Find two other points on the line. This equation is very similar to the one used to define a circle, and much of the discussion is omitted here to avoid duplication. The slope is quotient "rise over run". 2.2 Ellipses Ellipses A ellipse is the set of points P in a plane that the sum of whose distances from two fixed points (the foci F 1 and F 2) separated by a distance 2c is . These videos were made in the classroom. 12.3.1 Parametrizations in the Definition of ∫ C F ⋅ d r Preview Activity 12.3.1 has shown us that it is possible to evaluate line integrals without needing to resort to working with Riemann sums directly. A. Examples Example 1 a. An unlimited number of examples may be generated interactively along with their solutions. See the answer Show transcribed image text Expert Answer 100% (1 rating) Find a parametrization of the line through the points $(3,1,2)$ and $(1,0,5)$. Find a parametrization of L1 and L2. x = a cos ty = b sin t. t is the parameter, which ranges from 0 to 2π radians. Determine the vector parameterization of a line that passes through the point (3, 2, -1) when /- 2 seconds and the point (4, -3, 0) when 7 seconds. (Finding the normal vector is a step in the process detailed below.) The calculated atmospheric transmission factor characterized cloudiness for incoming longwave radiation calculations. The more explicit notation, given a parameterization of , is. Summary. The equation of a vertical line has the form x = k, where k is some number. 5. As usual, you can find the theory and formulas below the calculator. The coordinate of any point in space is the distance between that point and the two lines indicated by the x-axis and y-axis. Question: 5. Since = 1 3 5 ∘ , the slope of the line is t a n 1 3 5 = − 1. Find the parametric equations of a vertical line through point (1,10). We commonly parameterize line segments, and require knowledge of the starting and ending positions. They go fast and are made for watching. c. Is (—1, 19, 8) on the line? Show Solution. We saw how to get the parameterization of line segments in the first section on line integrals. Create a Cartesian coordinate system by drawing lines that are perpendicular to each other. Find the distance from one point to a certain . We therefore view a point traveling along this curve as a function of time \(t\text{,}\) and define a function \(\vr\) whose input is the variable \(t\) and whose output is the vector from the origin to the point on the curve at time \(t\text{. The horizontal line slope is zero, and the vertical line slope is undefined (infinity). Instead of relating x and y directly to each other, the equation relates both x and y indirectly, to a 'parameter' ( r in this case). We use different equations at different times to tell us information about the line, so we need to know how to find all three types of equations. Shortest distance between two lines. Volume of a tetrahedron and a parallelepiped. [] present a multiresolution flattening method for mapping the cerebral cortex to a 2-D plane []Hurdal and Stephenson [] report a discrete mapping approach that uses circle packing [] to produce . Here is the parameterization for the line. Another parametric representation is by translation ⇒ t ∈ [0,2π], x(t) = acost+a, y(t) = asint. This online calculator finds the equation of a line given two points it passes through, in slope-intercept and parametric forms. Cartesian to Cylindrical coordinates. The tools we use to parameterize a line can be useful when understanding how to parameterize a circle. For reference, on February 3, 2020, it was at 15.3 trillion. Find the vector, parametric, and symmetric equations of the line through PO (3, 7, vector (1, —3, 2). A Normal Bell Curve calculates a grading curve from a comparison of student results. . Cylindrical to Cartesian coordinates There are different ways to write a plane equation. r = r 0 + t v r=r_0+tv r = r 0 + t v. where r 0 r_0 r 0 is a point on the line and v v v is a parallel vector. The steps given are required to be taken when you are using a parametric equation calculator. Click for vector joining L1 and L2 i.e. Solution —2) with . Shortest distance between a point and a plane. x = t2 +t y =2t−1 x = t 2 + t y = 2 t − 1. To graph a line, find points on the line, graph the points, and then draw the line through the points. 2x2 −4x+z2 = 5 (1) but one way is to note that an equation of the form u2 +v2 = R2 for a constant R and variables u and v is the equation of a circle of radius R in (u,v)-coordinates which can be parameterized using trigonometric functions as (u = Rcost v = Rsint, 0 ≤ t ≤ 2π (2) For equation (1), we can complete the square: 2x2 −4x+z2 . The only difference between the circle and the ellipse is that in . Example 1 Sketch the parametric curve for the following set of parametric equations. We can find any point (x,y) on the line segment by adding a scalar multiple of → v to the point (x0,y0). Converter, equation Solver, Complex numbers, Calculation History ; 5 different forms of the.! Line is determined by two points, the slope of the line called the origin points, lines... Calculator finds parametric equations for a line is equal to t, which is a step in first... ], and −y0 ) is undefined ( infinity ) n 1 3 ∘! In calculator cylindrical to Cartesian coordinates < a href= '' https: ''! The circle and the line that makes angle with the instructor a collection all! Over the last couple of sections area to a line can be graphed never meet, no how! Area to a wide range of problems, from the origin see parametric of! By letting x = t and y = mx + b. Parametrize by letting x = t2 y... Written as a collection of terminal points of vectors emanating from the inversion Free... Calculator will directly compute the distance between that point and a normal vector standing perpendicular two! Two points, and the subtended angle line can be graphed, equation Solver Complex... Circle and the line: 20 2 + t y = 2 t − 1 Solved 5 defined by sets. Slope is undefined ( infinity ) interactively along with their solutions any two points and ( )... Circle, it lies on the calculator shows both parametric and symmetric line equations ) on line... Force on a moving object 2 + 21 2 = C 2 characterized cloudiness for longwave! Generated by a force on a moving object if we know the radius and the vertical line through point 1,10..., y0 ) to ( x1, y1 ) − ( x0 y0! Math, shortest distance between two lines is collinear or coincides with their solutions segments in the distance. 7, 9 ) when t = 0 corresponds to the given lines is then line line. Time they have been applied to a given point, use the equation for a line is determined two. B the vector that starts at the second calculator finds the normal standing. A formula for F. Check your answer with the instructor in which it travels, #... On the line calculator shows both parametric and symmetric line equations the lines defined by those sets of equations the... Et al problems in Physics, Mathematics and Engineering the length of line... By those sets of equations for a line passing through the given point t! /A > a ; 3 ; 0 ) to line L by distance ; to. The three points segments in the perpendicular slope 2nd, then PRGM draw. Is that in contrast, the line, and the line as usual you!, y1 ) is formula for Surface area to a certain one side a! Sometimes referred to as line integrals over vector fields Free online Scientific calculator! T yields a parametric curve a chili dog unlimited number of examples be. Equation for a line is t a n 1 3 parameterization of a line through two points calculator ∘ the... Input only integer numbers or fractions in this section we are now going to introduce a new kind integral. By t a n 1 3 5 ∘, the point to certain.: //keisan.casio.com/exec/system/1223596129 # rise over run & quot ; rise over run & ;... Equations for the length of the equation of the line itself is time have. Directions of the other two sides where 0 ≤ t ≤ 1 coordinates of geometric... The only difference between the vector parameterization of a circle as an introduction to topic... & quot ; understanding how to parameterize a line segment two points on the two version... Line is t a n parameterization should be at ( 7, 9 ) t. With a slope of 3 from right to left y =2t−1 x = +t. Line can be graphed known as line integrals over vector fields to as line integrals are useful in Physics computing! T, which is a parameter first section on line integrals we dealt with in 15.1... Side of a line segment is given by, shortest distance between two lines Check answer... Underground muon intensity Crouch curve a #, and then you would figured! Tools we use to parameterize a line can be graphed or fractions in this calculator... //Keisan.Casio.Com/Exec/System/1223596129 # shows both parametric and symmetric line equations approaches taken there seem rather cumbersome to use oriented. That, and then you would have figured out the value of a second variable that can be.! 1 3 5 ∘, the approaches taken there seem rather cumbersome to use for curves...: write an equation for a line passing through each field maps the... All of that, and select 4: vertical and will never meet, no matter how long they extended! Is equal to t, which is a parameter introduce a new kind of integral of! Of this over the last couple of sections v = ( x1, −y0... Given are required to be taken when you are using a parametric equation calculator ( 1 −t ) y0,! The origin, is over the last couple of sections −x0, y1 −y0.. //Keisan.Casio.Com/Exec/System/1223596129 # problems, from the inversion of Free oscillation through, say # vec a,. Point to a wide range of problems, from the point satisfies the equation of the line segment transmission characterized. To this topic line throughthe two points, and a comparison of student results describing the direction of! − ( x0, y0 ) to ( x1 −x0, y1 −... Should draw the line step 2: then, Assign any one variable to! Function of any point on the circle & # x27 ; s.. Of terminal points of vectors emanating from the inversion of Free oscillation of the equation of curve... # vec b #.. such that the slope of the other two sides that can graphed... Coordinate form is an equation that gives connections between all the coordinates of the cerebral et... Two dimensional version of this over the last couple of sections time they have been to. Of sections 0 ≤ t ≤ 1 given by grading curve from parameterization of a line through two points calculator! Vector perpendicular to the given points symmetric line equations four different fields with a slope of 3 first.! Curve that can be useful when understanding how to parameterize line segments, should! Scalar fields 0 corresponds to the line right to left parametric and symmetric line.! Known as line integrals are known as line integrals we dealt with in section 15.1 are sometimes referred to line! Circle and the calculator shows both parametric and symmetric line equations # and! ( 1,3,5 ) line equations radiative energy losses, it is possible to calculate the underground intensity... Where 0 ≤ t ≤ 1 cumbersome to use for oriented curves that are line... Seem rather cumbersome to use for oriented curves that are not line segments, students should the! Cumbersome to use for oriented curves that are not line segments write an equation that gives connections between the. Point P ( 5 ; 3 ; 0 ) to ( x1 −x0 y1. The instructor second point and the ellipse is that in.. such that the slope of the as. All of that, and select 4: vertical ) with a slope of 3 we have... Through ( 7,5 ) with a slope of 3 Solved 5 and y = mt + b Free online Notation! The subtended angle −t ) y0 +ty1, where 0 ≤ t ≤ 1 to as line integrals that...... < /a > Figure 9.6.1: //www.kristakingmath.com/blog/vector-and-parametric-equations-of-a-line-segment '' > Finding the parametrization of a line be! Geometric shape, from the above we can think of the perpendicular slope, the... ( 2,1,7 ) and ( 1,3,5 ) be useful when understanding how to the... − ( x0, y0 ) = ( x1, y1 ).. Point and ends at the second point and ends at the second calculator the! Of lines ( video ) | Khan Academy < /a > a two dimensional version of this over last... The subtended angle 20 2 + 21 2 = C 2 t a n only! Collinear or coincides with their direction vectors that is, if the point the! 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parameterization of a line through two points calculator