Table of Contents
- 1 How do you find orthogonal trajectory?
- 2 What is the orthogonal trajectory of parabola?
- 3 What is orthogonal trajectory in mathematics?
- 4 What is orthogonal trajectory with example?
- 5 What is the orthogonal trajectories of the family curve y 2 4ax?
- 6 What are orthogonal trajectories?
- 7 How do you find the perpendicular slope of an orthogonal family?
How do you find orthogonal trajectory?
Procedure to find orthogonal trajectory
- Let f(x,y,c)=0 be the equation of the given family of curves, where c is an arbitrary parameter.
- Differentiate f=0; w.r.t. ‘x’ and eliminate c,ie, form a differential equation.
- Substitute −dydx for dxdy in the above differential equation.
What is orthogonal trajectory of a curve?
The orthogonal trajectories are the curves that are perpendicular to the family everywhere. In other words, the orthogonal trajectories are another family of curves in which each curve is perpendicular to the curves in original family.
What is the orthogonal trajectory of parabola?
orthogonal trajectory, family of curves that intersect another family of curves at right angles (orthogonal; see figure). Solving this for the orthogonal curve gives the solution y2 + (x2/2) = k, which represents a family of ellipses (shown in red in the figure) orthogonal to the family of parabolas.
How do you find orthogonal trajectory in polar form?
Orthogonal Trajectories in Polar Co-ordinates If φ and φ’ denote the angles which the tangent to the given curve and the trajectory at the point of intersection (r, θ), make with the radius vector to the common point, φ ~ φ’ = (π/2) and so tanφ = – cotφ’. − 1 r d r d θ f o r r d θ d r i .
What is orthogonal trajectory in mathematics?
In mathematics an orthogonal trajectory is a curve, which intersects any curve of a given pencil of (planar) curves orthogonally. For example, the orthogonal trajectories of a pencil of concentric circles are the lines through their common center (see diagram).
How do you solve orthogonal?
Definition. Two vectors x , y in R n are orthogonal or perpendicular if x · y = 0. Notation: x ⊥ y means x · y = 0. Since 0 · x = 0 for any vector x , the zero vector is orthogonal to every vector in R n .
What is orthogonal trajectory with example?
What is oblique trajectory?
OBLIQUE TRAJECTORIES A curve which intersects the curves of the given family at a constant angle α ≠ 90º is called an oblique trajectory of the given family.
What is the orthogonal trajectories of the family curve y 2 4ax?
The given curve isy2 = 4ax i2ydy/dx = 4aa = ydy/2dx iiEliminating a from Eqs i and ii we get y2 = 4xydy/2dxy = 2xdy/dxReplacing dy/dx by – dx/dy we gety = 2x x -dy/dxydy + 2xdx = 0Integrating we get y2/2 + x2 = cwhich is the required orthogonal trajectory.
What is orthogonal in maths?
Orthogonal is commonly used in mathematics, geometry, statistics, and software engineering. Most generally, it’s used to describe things that have rectangular or right-angled elements. More technically, in the context of vectors and functions, orthogonal means “having a product equal to zero.”
What are orthogonal trajectories?
The orthogonal trajectories are the curves that are perpendicular to the family everywhere. In other words, the orthogonal trajectories are another family of curves in which each curve is perpendicular to the curves in original family.
What does it mean when two curves are orthogonal?
Two curves are orthogonal or perpendicular at a point if their respective tangent lines to the curves at that point are perpendicular. The word orthogonal comes from the Greek ορθη (right) and γωνια (angle); the word trajectory comes from the Latin trajectus (cut across).
How do you find the perpendicular slope of an orthogonal family?
If we want to find the orthogonal trajectories, and we know that they’re perpendicular to our family everywhere, then we want a slope for the orthogonal trajectories that is perpendicular to the slope of the original family. To find a perpendicular slope, we take the negative reciprocal (flip it upside down and add a negative sign).