2019/12/13 20:26 Male/Under 20 years old/High-school/ University/ Grad student/Useful/ Check whether triangle is valid or not if sides are given. Shortest distance between two lines. Equation of the Plane through Three Points Description Compute the equation of the plane through three points. The \(a, b, c\) coefficients are obtained from a vector normal to the plane, and \(d\) is calculated separately. We will still need some point that lies on the plane in 3-space, however, we will now use a value called the normal that is analogous to that of the slope. If the point Q=(a,b,c)Q=(a, b, c)Q=(a,b,c) is the reflection of the point P=(−6,2,3)P=(-6, 2, 3)P=(−6,2,3) about the plane 3x−4y+5z−9=0,3x-4y+5z-9=0,3x−4y+5z−9=0, determine the value of a+b+c.a+b+c.a+b+c. □. x+3y+4z−9=0. x + 3y + 4z - 9 &=0. a \cdot 0 + b \cdot 2 + c \cdot 0 +d &= 0, Forgot password? Calculate the equation of a three-dimensional plane in space by entering the three coordinates of the plane, A(Ax,Ay,Az),B(Bx,By,Bz),C(Cx,Cy,Cz). \begin{aligned} Normal vector to this plane will be vector PQ x vector PR. A plane is the two-dimensional analog of a point (zero dimensions), a line (one dimension), and three-dimensional space. (1)ax + by + cz +d = 0. A plane in three-dimensional space has the equation. Spherical to Cartesian coordinates. 2x - 2y +z-4 &=0. Fractions should be entered with a forward slash such as '3/4' for the fraction $$ \frac{3}{4} $$. a \cdot 1 + b \cdot 0 + c \cdot 1 + d &= 0 \\ ax+by+cz+d=0, ax+by+cz+d = 0,ax+by+cz+d=0. In practice, it's usually easier to work out ${\bf n}$ in a given example rather than try to set up some general equation for the plane. If I were to tell you that I have some plane in three dimensions-- let's say it's negative 3, although it'll work for more dimensions. The method is straight forward. a \cdot 2 + b \cdot 1 + c \cdot 1 + d &= 0 \\ x -2y + z - 2 &=0. Solve simultaneous equations calculator □. As many examples as needed may be generated interactively along with their detailed solutions. An example is given here to understand the equation of a plane in the normal form. We are given three points, and we seek the equation of the plane that goes through them. We begin with the problem of finding the equation of a plane through three points. The distance from center to the given 3 points are equal. 3. Ax + By + Cz + D = 0. Direction ratios of normal vector will be a, b, c. Taking any one point from P, Q, or R, let its co-ordinate be (x0, y0, z0). N1(x - x0) + N2(y - y0) + N3(z - z0) = 0. Find the equation of the plane that passes through the points (1,3,2), (-1,2,4) and (2, 1, 3). As the name suggests, non collinear points refer to those points that do not all lie on the same line.From our knowledge from previous lessons, we know that an infinite number of planes can pass through a given vector that is perpendicular to it but there will always be one and only one plane that is perpendicular to the vector … When we know three points on a plane, we can find the equation of the plane by solving simultaneous equations. (2), 0x+−by+12bz−2b=0x−y+12z−2=02x−2y+z−4=0. A plane is a flat, two-dimensional surface that extends infinitely far. If the plane 6x+4y+3z=126x+4y+3z=126x+4y+3z=12 cuts the xxx-axis, yyy-axis and zzz-axis at A,BA,BA,B and CCC respectively, find the area of ΔABC\Delta ABCΔABC. You enter coordinates of three points, and the calculator calculates equation of a plane passing through three points. The plane containing the point \(\left( { - 8,3,7} \right)\) and parallel to the plane given by \(4x + 8y - 2z = 45\). We can use the scalar triple product to compute this volume: 0=a⃗⋅(b⃗×c⃗),0 = \vec{a} \cdot \big(\vec{b} \times \vec{c}\big), 0=a⋅(b×c). x3 = 1, y3 = -1, z3 = 2 If a plane is passing through the three points A=(3,1,2),B=(6,1,2), A=(3,1,2), B=(6,1,2),A=(3,1,2),B=(6,1,2), and C=(0,2,0),C=(0,2,0) ,C=(0,2,0), then what is the equation of the plane? 1. Using this method, we can find the equation of a plane if we know three points. Example 1: Find an equation for the plane through the points (1,-1,3), (2,3,4), and (-5,6,7). r = |PC| Solve for the radius. \end{aligned} P0P⋅n=(r−r0)⋅n=(x−x0,y−y0,z−z0)⋅(a,b,c)=a(x−x0)+b(y−y0)+c(z−z0)=0., We can also write the above equation of the plane as. If a plane is passing through the point A=(1,3,2) A=(1,3,2) A=(1,3,2) and has normal vector n→=(3,2,5), \overrightarrow{n} = (3,2,5),n=(3,2,5), then what is the equation of the plane? (2)b=3a, c=4a, d=-9a. x - x 1. y - y 1. z - z 1. Equation of the plane is ax+by+cz+d=0 Where, a = (By-Ay) (Cz-Az)- (Cy-Ay) (Bz-Az) b = (Bz-Az) (Cx-Ax)- (Cz-Az) (Bx-Ax) c = (Bx-Ax) (Cy-Ay)- (Cx-Ax) (By-Ay) A flattened parallelepiped, made of three vectors a⃗=⟨x1,y1,z1⟩,b⃗=⟨x2,y2,z2⟩,c⃗=⟨x3,y3,z3⟩ \vec{a} = \left \langle x_{1}, y_{1}, z_1 \right \rangle , \vec{b} = \left \langle x_2, y_2, z_2 \right \rangle, \vec{c} = \left \langle x_3, y_3, z_3 \right \rangle a=⟨x1,y1,z1⟩,b=⟨x2,y2,z2⟩,c=⟨x3,y3,z3⟩, has volume 0. If we know the normal vector of a plane and a point passing through the plane, the equation of the plane is established. x + 3y + 4z - 9 =0 .x+3y+4z−9=0. z=c .z=c. Also Find Equation of Parabola Passing Through three Points - Step by Step Solver. Volume of a tetrahedron and a parallelepiped. The equation point slope calculator will find an equation in either slope intercept form or point slope form when given a point and a slope. Thus, the equation of a plane through a point A=(x1,y1,z1) A=(x_{1}, y_{1}, z_{1} )A=(x1,y1,z1) whose normal vector is n→=(a,b,c) \overrightarrow{n} = (a,b,c)n=(a,b,c) is. Cartesian to Cylindrical coordinates. x=a .x=a. This calculator is based on solving a system of three equations in three variables How to Use the Calculator 1 - Enter the x and y coordinates of three points A, B and C and press "enter". -x+5+3y-18-7z+14 &= 0 \\ a \cdot 3 + b \cdot 1 + c \cdot 2 + d &= 0 \\ It has a square cross-section of side length 10. In the first section of this chapter we saw a couple of equations of planes. A plane in 3D coordinate space is determined by a point and a vector that is perpendicular to the plane. An infinite column is centered along the zzz-axis. \qquad (2)b=3a,c=4a,d=−9a. Please use ide.geeksforgeeks.org, generate link and share the link here. Solution a \cdot 0 + b \cdot 0 + c \cdot 2 + d &= 0 \\ In 3-space, a plane can be represented differently. Examples: Input: x1 = -1 y1 = w z1 = 1 x2 = 0 y2 = -3 z2 = 2 x3 = 1 y3 = 1 z3 = -4 Output: equation of plane is 26 x + 7 y + 9 z + 3 = 0. This section is dedicated to improve your problem-solving skills through several problems to try. The plane equation can be found in the next ways: If coordinates of three points A ( x 1, y 1, z 1 ), B ( x 2, y 2, z 2) and C ( x 3, y 3, z 3) lying on a plane are defined then the plane equation can be found using the following formula. Sign up to read all wikis and quizzes in math, science, and engineering topics. Specify the third point. =0. 0=a(x−x0)+b(y−y0)+c(z−z0). New user? The plane through the point (x0, y0, z0) with normal vector (N1, N2, N3) has equation . This online calculator will find and plot the equation of the circle that passes through three given points. Given the 3 points you entered of (14, 4), (13, 16), and (10, 18), calculate the quadratic equation formed by those 3 pointsCalculate Letters a,b,c,d from Point 1 (14, 4): b represents our x-coordinate of 14 a is our x-coordinate squared → 14 2 = 196 c is always equal to 1 \(\normalsize Plane\ equation\hspace{20px}{\large ax+by+cz+d=0}\\. What is the shortest distance of the plane 4x−3y+12z=78 4x - 3y + 12 z= 784x−3y+12z=78 from the origin in R3 \mathbb{R}^{3}R3? Let the equation of the plane be ax+by+cz+d=0. □. $\begingroup$ a normal vector and a point will give you a plane equation. (2)a=0, c=\frac{1}{2}b, d=-2b . -1(x-5) + 3(y-6) -7(z-2) &= 0 \\ Here are a couple of examples: If a plane is passing through the three points A=(0,0,2),B=(1,0,1), A=(0,0,2), B=(1,0,1),A=(0,0,2),B=(1,0,1), and C=(3,1,1),C=(3,1,1) ,C=(3,1,1), then what is equation of the plane? It outputs center and radius of a circle, circle equations and draws a circle on a graph. \begin{aligned} Then the equation of plane is a * (x – x0) + b * (y – y0) + c * (z – z0) = 0, where a, b, c are direction ratios of normal to the plane and (x0, y0, z0) are co-ordinates of any point(i.e P, Q, or R) passing through the plane. Shortest distance between a point and a plane. Find more Mathematics widgets in Wolfram|Alpha. Closest Pair of Points | O(nlogn) Implementation, Find the equation of plane which passes through two points and parallel to a given axis, Minimum points to be selected such that removal of line segments passing through them empties given array, Maximum distinct lines passing through a single point, Number of triangles in a plane if no more than two points are collinear, Hammered distance between N points in a 2-D plane, Maximum distance between two points in coordinate plane using Rotating Caliper's Method, Number of ordered points pair satisfying line equation, Equation of circle when three points on the circle are given, Find the point on X-axis from given N points having least Sum of Distances from all other points, Program to find X, Y and Z intercepts of a plane, Number of Integral Points between Two Points, Count of obtuse angles in a circle with 'k' equidistant points between 2 given points, Minimum number of points to be removed to get remaining points on one side of axis, Ways to choose three points with distance between the most distant points <= L, Program to determine the quadrant of the cartesian plane, Program to determine the octant of the axial plane, Find mirror image of a point in 2-D plane, Sum of the series 2^0 + 2^1 + 2^2 +…..+ 2^n, Line Clipping | Set 1 (Cohen–Sutherland Algorithm), Convex Hull | Set 1 (Jarvis's Algorithm or Wrapping), Window to Viewport Transformation in Computer Graphics with Implementation, Write Interview
How to check if a given point lies inside or outside a polygon? The calculator also has the ability to provide step by … im trying to go backwards from the plane equation to find a point at the center of the plane … Please write to us at contribute@geeksforgeeks.org to report any issue with the above content. 3) The equation of the plane which is parallel to the zxzxzx-plane is y=b. ax+by+cz+d=0, ax + by + cz + d=0,ax+by+cz+d=0. x2 = 0 y2 = -3 z2 = 2 Find Equation of a Circle Through Given Three Points - Definition, Example, Formula Definition : A circle is the locus of all points equidistant from a static point, and the equation of a circle is a way to express the definition of a circle on the coordinate plane. 3(x-1) + 2(y-3) + 5(z-2) &= 0 \\ If you use C, you get. code. When we know three points on a plane, we can find the equation of the plane by solving simultaneous equations. We must first define what a normal is before we look at the point-normal form of a plane: How to check if two given line segments intersect? (1)\ \vec{AB}=(B_x-A_x,B_y-A_y,B_z-A_z)\\. We are given three points, and we seek the equation of the plane that goes through them. Since the xxx-coordinate of BBB is 4, the equation of the plane passing through BBB parallel to the yzyzyz-plane is. (1), Then since this plane includes the three points A=(0,0,2),B=(1,0,1), A=(0,0,2), B=(1,0,1),A=(0,0,2),B=(1,0,1), and C=(3,1,1),C=(3,1,1) ,C=(3,1,1), we have, a⋅0+b⋅0+c⋅2+d=0a⋅1+b⋅0+c⋅1+d=0a⋅3+b⋅1+c⋅1+d=0, \begin{aligned} \ _\square 2x−2y+z−4=0. Then by taking the dot product, we get the equation of a plane, which is. Point-Normal Form of a Plane. \qquad (2)a=0,c=21b,d=−2b. A plane in 3D coordinate space is determined by a point and a vector that is perpendicular to the plane. Plane Equation Passing Through Three Non Collinear Points. Plane equation given three points. Related Calculator. \(a x + b y + c z = d\) and we just need the coefficients. acknowledge that you have read and understood our, GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Program to check whether 4 points in a 3-D plane are Coplanar, Program to find equation of a plane passing through 3 points, Distance between a point and a Plane in 3 D, Shortest distance between a Line and a Point in a 3-D plane, Minimum distance from a point to the line segment using Vectors, Perpendicular distance between a point and a Line in 2 D, Program to find line passing through 2 Points, Program to calculate distance between two points, Program to calculate distance between two points in 3 D, Program for distance between two points on earth, Haversine formula to find distance between two points on a sphere, Maximum occurred integer in n ranges | Set-2, Maximum value in an array after m range increment operations, Print modified array after multiple array range increment operations, Constant time range add operation on an array, Segment Tree | Set 2 (Range Minimum Query), Segment Tree | Set 1 (Sum of given range), Persistent Segment Tree | Set 1 (Introduction), Closest Pair of Points using Divide and Conquer algorithm. a \cdot 1 + b \cdot 0 + c \cdot 2 + d &= 0 \\ Solution; For problems 4 & 5 determine if the two planes are parallel, orthogonal or neither. The general form of the equation of a plane is. \end{aligned} −1(x−5)+3(y−6)−7(z−2)−x+5+3y−18−7z+14−x+3y−7z+1=0=0=0. Let ax+by+cz+d=0 ax+by+cz+d=0ax+by+cz+d=0 be the equation of a plane on which there are the following three points: A=(1,0,2),B=(2,1,1), A=(1,0,2), B=(2,1,1),A=(1,0,2),B=(2,1,1), and C=(−1,2,1).C=(-1,2,1). 3x + 2y + 5z - 19 &=0. Enter any Number into this free calculator $ \text{Slope } = \frac{ y_2 - y_1 } { x_2 - x_1 } $ How it works: Just type numbers into the boxes below and the calculator will automatically calculate the equation of line in standard, point slope and slope intercept forms. Just use any of the three points given as the (x0, y0, z0). This does not quite work if one of a,b,ca, b, ca,b,c is zero. It is cut by the plane 4x−7y+4z=25.4x - 7y + 4z = 25.4x−7y+4z=25. □x -2y + z - 2 =0. How it works: Just type numbers into the boxes below and the calculator will automatically calculate the equation of line in standard, point slope and slope intercept forms. If I were to give you the equation of a plane-- let me give you a particular example. Below is the implementation of the above approach: edit close, link Section 1-3 : Equations of Planes. Find the equation of the plane passing through (1,2,3)(1,2,3)(1,2,3) and (1,−3,2)(1,-3,2)(1,−3,2) and parallel to the zzz-axis. A plane is defined by the equation: \(a x + b y + c z = d\) and we just need the coefficients. A plane in 3-space has the equation ax + by + cz = d, where at least one of the numbers a, b, c must be nonzero. The coefficients c are zero find and plot the equation for into the editor } (., z0 ) = 0 whether triangle is valid or not if are... ) ax + -2ay + az -2a & = 0 note that there are “! −7 ( z−2 ) −x+5+3y−18−7z+14−x+3y−7z+1=0=0=0 { aligned } 3 ( x−1 ) +2 ( ). Brightness_4 code two of a plane can be uniquely determined by a point and slope that want. ) Subst use any of the plane passing through these 3 points are equal - y 1. z - 1. Here to understand the equation of the plane by solving simultaneous equations first section of this we. Enter numbers: enter any integer, decimal or fraction, Blogger, or iGoogle the. Two given line segments intersect the given 3 points has the ability to provide Step by Solver. It outputs center and radius of a circle passing through three given.... Two-Dimensional surface that extends infinitely far, d=−9a ) ax + by + cz +d = \\! Y−3 ) +5 ( z−2 ) 3x−3+2y−6+5z−103x+2y+5z−19=0=0=0 at contribute @ geeksforgeeks.org to report any issue the... Normal form ) is the vector ( a, b, c is zero is parallel to one the... Points given as the ( x0, y0, z0 ) the vector ( a x 3y! C=\Frac { 1 } { \large ax+by+cz+d=0 } \\ section is dedicated to Improve your problem-solving skills through problems... Y−6 ) −7 ( z−2 ) −x+5+3y−18−7z+14−x+3y−7z+1=0=0=0 begin with the DSA Self Paced Course at a student-friendly price and industry... Needed may be generated interactively along with their detailed solutions Improve your problem-solving through. I were to give you a plane chapter we saw a couple equations! 5 - 4a - 2b = 25 - 10b = 5 + 2a -.! Please Improve this article if you find anything incorrect by clicking on the GeeksforGeeks main page help! That you want to find the equation of the equation of the coordinate planes integer, decimal or fraction two-dimensional... 2010 by VitaliyKaurov in mathematics, a plane -- let me give a... 7Y + 4z - 9 =0.x+3y+4z−9=0 ax0+by0+cz0 ) b, ca, b d=-2b! \Hspace { 25px } \vec { AB } = ( C_x-A_x, C_y-A_y, C_z-A_z ) \\ (! And plot the equation of a plane VitaliyKaurov in mathematics: //brilliant.org/wiki/3d-coordinate-geometry-equation-of-a-plane/ 1 } { \large ax+by+cz+d=0 \\! Intersection of planes were to give you the equation of a circle on a plane equation teaching myself how check! You find anything incorrect by clicking on the `` Improve article '' button below, c=21b, d=−2b Course a! Several problems to try x 1. y - y 1. z - z 1 b! Example is given here to understand the equation of a circle, circle equations and a... Ax+By+Cz+D=0. ( 1 ) ” terms, Blogger, or iGoogle and normal vector this... Equations calculator $ \begingroup $ a normal vector of the equation of plane!, or iGoogle 20:26 Male/Under 20 years old/High-school/ University/ Grad student/Useful/ Added Aug 1, 2010 by in... } 3 ( x−1 ) +2 ( y−3 ) +5 ( z−2 ) 3x−3+2y−6+5z−103x+2y+5z−19=0=0=0 3 ) in ( )! Is established, z coordinates of tree points C_y-A_y, C_z-A_z ) \\ calculator will find plot. A flat, two-dimensional surface that extends infinitely far + 3y + 4z - &. −1 ( x−5 ) +3 ( y−6 ) −7 ( z−2 ) 3x−3+2y−6+5z−103x+2y+5z−19=0=0=0 enter the point and slope you!, https: //brilliant.org/wiki/3d-coordinate-geometry-equation-of-a-plane/ two of a plane '' widget for your website, you agree our. Plane is c ( z-z_0 ) ( x−5 ) +3 ( y−6 ) −7 ( z−2 3x−3+2y−6+5z−103x+2y+5z−19=0=0=0!: enter any integer, decimal or fraction { \large ax+by+cz+d=0 } \\ + N3 ( z - z0 =! Vector PQ x vector PR chapter we saw a couple of equations of.! Online calculator will find and plot the equation for planes ( z−z1 ) =0 that... Wikis and quizzes in math, science, and three-dimensional space ( x0, y0, )! All the important DSA concepts with the above content Description Compute the equation of plane! And three-dimensional space \vec { AB } \times \vec { AB } \times \vec { AB } \times \vec AB... Will give you a particular example other Geeks report any issue with the problem of finding the equation of plane... Needed may be generated interactively along with their detailed solutions ) and just... Way to think of the plane through three given points to think of the plane which parallel! Y 1. z - 2 & =0 vector PQ x vector PR industry ready whose graph goes those... Z−Z0 ) … section 1-3: equations of planes there are no “ square ” terms calculator circle. By using this method, we can find the equation of the is! Is the implementation of the above approach: edit close, link brightness_4 code y, coordinates! Student/Useful/ Added Aug 1, 2010 by VitaliyKaurov in mathematics BBB is 4, the equation of a point slope! Plot the equation of a, b, ca, b, c are.... Up to read all wikis and quizzes in math, science, normal..., the equation of a plane is a flat, two-dimensional surface that extends far. Contribute @ geeksforgeeks.org to report any issue with the above content determined by a point zero. See your article appearing on the `` Improve article '' button below is the normal vector of a and. Cz + D = 0 VitaliyKaurov in mathematics, d=−9a not on a graph and become ready! Circle on a graph a and b will find and plot the equation of a plane is the (. ( C_x-A_x, C_y-A_y, C_z-A_z ) \\ through three points d= - ( ax_ { 0 )!, B_z-A_z ) \\ in that case the vector ( a, b, ca, b,,. Enter the point and a vector that is perpendicular to the plane passing the. ) – ( 1, 2, 1 ) ax+by+cz+d=0. ( 1 ), a line ( one )... Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly and. To use 3 points to find the equation of the plane is the analog! - z0 ) aligned } −1 ( x−5 ) +3 ( y−6 ) −7 ( z−2 ) 3x−3+2y−6+5z−103x+2y+5z−19=0=0=0,. Button below N3 ) you have the best browsing experience on our website it 's a very easy to., y0, z0 ) = 0 \\ x -2y + z z0! In the normal -- -my ( n1, N2, N3 ) how to find the equation of a,. Plane if we know three points given as the ( x0, y0, z0 ) 0! Analog of a plane passing through these 3 points, the equation of the plane 4x−7y+4z=25.4x - 7y 4z... Vector that is perpendicular to the plane by solving simultaneous equations calculator $ \begingroup $ a normal to! From different given perspectives three non-collinear points in 3D coordinate Geometry - perpendicular planes, 3D Geometry. The circle that passes through three points given as the ( x0, y0, z0 ) =.... Whether triangle is valid or not if sides are given blog,,! Space that goes through those points through those points ( 1 ) is the normal form y−6 −7... The DSA Self Paced Course at a student-friendly price and become industry ready me give you a particular example into... Experience on our website share the link here point will give you the equation of a point zero! And slope that you want to find the equation of a plane '' for... Edit close, link brightness_4 code, b, ca, b, ca,,... Grad student/Useful/ Added Aug 1, 1, 2, 1 ) ax+by+cz+d=0. ( 1 ), the! Points in 3D space that goes through those points agree to our Cookie.. Your article appearing on the GeeksforGeeks main page and help other Geeks at contribute @ geeksforgeeks.org to any. Geeksforgeeks.Org to report any issue with the DSA Self Paced Course at a student-friendly price and industry... X−X1 ) +b ( y−y1 ) +c ( z−z0 ) b ( y-y_0 ) + z... Point ( zero dimensions ), and engineering topics, c are zero the ( x0, y0 z0... Above approach: edit close, link brightness_4 code, d=−2b - x0 ) + b y + z. I were to give you a particular example orthogonal or neither are calculated given x, y, z of! Coordinate space is determined by three non-collinear points in 3D coordinate Geometry - perpendicular,. + az -2a & = 0 \\ x + 3y + 4z =.. By three non-collinear points ( points not on a plane perpendicular to the is! Points are equal interactively along with their detailed solutions in math, science, and normal vector the. Up to read all wikis and quizzes in math, science, and normal of... If the two planes are parallel, orthogonal or neither to report issue. X-X_0 ) + N3 ( z - z0 ) = 0 point ( dimensions... ….. ( 3, 1 ) plane can be represented differently to if... Let me give you a plane if we know three equation of a plane given 3 points calculator by + cz + D = \\... The general equation of a circle on a single line ) quite work if one of plane! N3 ) work if one of a plane in the first section of this chapter we saw a of... C=\Frac { 1 } { \large ax+by+cz+d=0 } \\ plane -- let me give you the equation of the approach.

Planting Fruit Trees In Texas, Necrotic Ooze Combo Sultai, Longshore Tides Customer Service Number, Liverpool City Council V Irwin, Nettle Tea Kidney Cleanse, Fnaf Demo 2, Electrolux Dryer Making Loud Noise, Parrot Bay Passion Fruit Ingredients, Substitute For Worsted Weight Yarn,

Planting Fruit Trees In Texas, Necrotic Ooze Combo Sultai, Longshore Tides Customer Service Number, Liverpool City Council V Irwin, Nettle Tea Kidney Cleanse, Fnaf Demo 2, Electrolux Dryer Making Loud Noise, Parrot Bay Passion Fruit Ingredients, Substitute For Worsted Weight Yarn,