Mathematics
Equation of Line in 3D
The equation of a line in 3D is represented by a vector equation or parametric equations. The vector equation involves a point on the line and a direction vector, while the parametric equations involve three equations that represent the x, y, and z coordinates of points on the line in terms of a parameter t.
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4 Key excerpts on "Equation of Line in 3D"
eBook - ePub- Nita H. Shah, Falguni S. Acharya(Authors)
- 2023(Publication Date)
- River Publishers(Publisher)
2 Straight Line 2.1 Representation of Line (Introduction) A straight line may be generated by the intersection of two non-parallel planes. So, the general equation of a straight line is given by the combined equations of two planes as given by a 1 x + b 1 y + c 1 z + d 1 = 0 a n d a 2 x + b 2 y + c 2 z + d 2 = 0 ; parovided a 1 a 2 ≠ b 1 b 2 ≠ c 1 c 2. Remark: A straight line is also called a right line. The x axis is the intersection of the XZ and XY planes, y = 0 and z = 0 taken together are its equation. Similarly, x = 0, z = 0 are the equations of the y axis, and x = 0, y = 0 are the equations of the z axis. 2.2 Equation of a Straight Line in the Symmetrical Form To find the equations of the line passing through a given point A (x 1, y 1, z 1) and having direction cosines l, m, n: Let A (x 1, y 1, z 1)be the fixed point on a straight line and P (x, y, z) be any given point on the given line and let AP = r. Let l, m, n be the direction cosines of this line. Assuming projections of AP on the coordinate axes, we get (2.1) x − x 1 = l r ; y − y 1 = m r a n d z − z 1 = n r ∴ x − x 1 l = y − y 1 m = r which is the equation of a straight line in symmetrical form. The coordinates (x, y, z) of any points on the line is given by x = x 1 ± lr ; y = y 1 ± mr and z = z 1 + nr i.e., (x 1 + lr, y 1 + mr, z 1 + nr) ; r ϵ. ℝ If l ≠ 0, m ≠ 0, n ≠ 0 or equivalently lmn ≠ 0, x − x 1 l = y − y 1 m = z − z 1 n are the two required equations of the line. Remark: The equation of the coordinate axes in symmetrical form can be represented respectively by x − 0 1 = y − 0 0 = z − 0 0 ; x − 0 0 = y − 0 1 = z − 0 0 ; x − 0 0 = y − 0 0 = z − 0 1. (0, 0,
eBook - PDF- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
Thus the equation of the line passing through (1, 2, −1) with direction ratios ⟨0, −2, 6⟩ is given by x − 1 0 = y − 2 −2 = z + 1 6 = t, for t ∈ ℝ Vector Equations of Lines We will now show how vector notation can be used to express the parametric equations of a line more compactly. Because two vectors are equal if and only if their components are equal, (1) and (2) can be written in vector form as ⟨x, y⟩ = ⟨x 0 + at, y 0 + bt⟩ ⟨x, y, z⟩ = ⟨x 0 + at, y 0 + bt, z 0 + ct⟩ or, equivalently, as ⟨x, y⟩ = ⟨x 0 , y 0 ⟩ + t⟨a, b⟩ (5) ⟨x, y, z⟩ = ⟨x 0 , y 0 , z 0 ⟩ + t⟨a, b, c⟩ (6) For the equation in 2-space we define the vectors r, r 0 , and v as r = ⟨x, y⟩ , r 0 = ⟨x 0 , y 0 ⟩ , v = ⟨a, b⟩ (7) and for the equation in 3-space we define them as r = ⟨x, y, z⟩ , r 0 = ⟨x 0 , y 0 , z 0 ⟩ , v = ⟨a, b, c⟩ (8) Substituting (7) and (8) in (5) and (6), respectively, yields the equation r = r 0 + tv (9) in both cases. We call this the vector equation of a line in 2-space or 3-space. In this equation, v is a nonzero vector parallel to the line, and r 0 is a vector whose components are the coordinates of a point on the line. P 0 r 0 v r t v t v L x y r = r 0 + t v ▴ Figure 11.5.4 We can interpret Equation (9) geometrically by positioning the vectors r 0 and v with their initial points at the origin and the vector tv with its initial point at P 0 (Figure 11.5.4). The vector tv is a scalar multiple of v and hence is parallel to v and L. Moreover, since the initial point of tv is at the point P 0 on L, this vector actually runs along L; hence, the vector r = r 0 + tv can be interpreted as the vector from the origin to a point on L. As the parameter t varies from 0 to +∞, the terminal point of r traces out the portion of L that extends from P 0 in the direction of v, and as t varies from 0 to −∞, the terminal point of r traces out the portion of L that extends from P 0 in the direction that is opposite to v.
- Carl D. Crane, III, Michael Griffis, Joseph Duffy(Authors)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
1 Geometry of Points, Lines, and Planes . . . and then geometry will become what geometry ought to be. Mr. Querulous Ball’s “A Dynamical Parable” (1887) 1.1 Introduction Points, lines, and planes are the fundamental elements of spatial geometry. A point can be thought of as a location in 3D space, and its coordinates have units of length. A line can be considered to be an infinite collection of points defined by a direction (which is a dimensionless vector) that passes through some given point (which has units of meters). A plane is a two-dimensional set of points that can be defined, for example, by three points or by a line and one point. This chapter introduces the concept of homogeneous coordinates as applied to points, lines, and planes. The homogeneous coordinates of each will be defined together with the equation for each. The equation of a point, line, or plane will be shown to be a vector equation where any vector that satisfies that equation is a member of that point, line, or plane. 1.2 The Position Vector of a Point The position vector to a point Q 1 from a reference point O will be referred to as r 1 and can be expressed in the form r 1 = x 1 i + y 1 j + z 1 k w 1 (1.1) or r 1 w 1 = S O1 , (1.2) where S O1 = x 1 i + y 1 j + z 1 k and the components of the vector S O1 have units of length. The term w 1 is dimensionless. In Figure 1.1 it is assumed that w 1 = 1 and (x 1 , y 1 , z 1 ) are the usual Cartesian coordinates for the point Q 1 . The coordinates r of some general point Q may be expressed as 2 1 Geometry of Points, Lines, and Planes Figure 1.1 Coordinates of a point r w = S O , (1.3) where S O = x i + y j + z k. The subscript O has been introduced to signify that S O is origin dependent. Clearly, if we choose some other reference point, the actual point Q would not change. However, the coordinates (x , y , z), which determine Q, would change. The ratios x/w, y/w, and z/w are three independent scalars and, therefore, there are ∞ 3 points in space.
eBook - PDFA Collection of Problems in Analytical Geometry
Three-Dimensional Analytical Geometry
- D. V. Kletenik, W. J. Langford, E. A. Maxwell(Authors)
- 2016(Publication Date)
- Pergamon(Publisher)
1002. Find an equation for the plane belonging to the pencil of planes given by +3z —5 =0. 1006. Set up equations for the projection of the line given by 5x-4y-2z-5 = 0 x + 2z-2 = 0 on to the plane given by 2x— y+z — 1 = 0. § 42. The Direction-Vector of a Straight Line. The Canonical Equations of a Straight Line. Parametric Equations of a Straight Line Any non-zero vector lying on a given straight line or parallel to it is called a direction-vector of the straight line. A direction-vector of an arbitrary straight line will be denoted by a and its coordinates by /, m, n: a = {/, m, n). A straight line through the point M 0 (x 0 , y 0 , z 0 ) and having the direction-vector a = {/, m, n} is definable by the two equations I m (1) these are called the canonical equations of a straight line. The canonical equations of a straight line through two given points M 1 (x l9 y l9 z x ), M 2 (x 2 , J>2> z 2) n a v e t n e form *~-*i _ y-y± _ z-zi ( 2 ) x^~ x i y%~yi z 2 ~ z i If each of the equal ratios in the canonical equations is put equal to t, we obtain 66 THREE-DIMENSIONAL ANALYTICAL GEOMETRY X = X 0 + It Ί y = Jo + mt ' ( 3 ) z = z 0 --nt J These are the parametric equations of the straight line which goes through the point M 0 (x 0 , y 0 , z 0 ) in the direction of the vector a = {/, m, n).
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