Website owner:  James Miller

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Def. Position vector (or radius vector). The vector


running from the origin to a point P(x, y) in the plane is called the position vector or radius vector of P. The vectors ole2.gif and ole3.gif are unit vectors along the positive the x and y axes respectively.

Plane curves. We can think of a curve in the plane as a path of a moving point. The definition of a plane curve is essentially an analytical implementation of this view. A plane curve is defined parametrically as the graph of parametric equations

1)        x = x(t)

            y = y(t)

where the functions x(t) and y(t) are continuous and the range of the parameter t is some interval (finite or infinite) of the real axis. In vector language a plane curve is defined as the graph of points traced out by a position vector


where the functions f and g are continuous and the range of the parameter t is some interval (finite or infinite) of the real axis. The positive direction of a space curve is the direction of increasing t. i.e Δs/Δt > 0.

If we interpret t as time, 1) can be regarded as defining the path of a moving point. The point may pass through the same point in the plane several times which means that the curve may intersect itself. This definition that we have given gives a curve that is very general and one that may not be very smooth. For example, it could include things like the track of a tiny particle in Brownian movement over a long period of time (a very random path going first in one direction and then abruptly changing to another direction — randomly and suddenly changing from one direction to another).

Derivative of a vector. The derivative of a vector




with respect to t is given by



The derivative ole8.gif is a vector tangent to the curve at the point in question. See Fig. 2. If the variable t represents time, then ole9.gif represents the velocity with which the terminal point of the radius vector describes the curve. Similarly, dv/dt represents its acceleration a along the curve.

Unit tangent vector. Let a curve C be given by

            R(s) = x(s) i + y(s) j

where the parameter s represents the arc length measured from some fixed point on the curve. Then the unit tangent vector to the curve at a particular point P is given by


            T = dR/ds 


That this is so can be seen from Fig. 3 which shows R and R + ΔR at points P and P'. The quotient ΔR/Δs is a vector along the line of the chord PP'. Since the length of ΔR is the length of the chord PP', we see that when P' approaches P the limit of the length of ΔR/Δs is unity. Furthermore, the limiting direction of PP' is that of the tangent at P. Therefore



Differentiation of R with respect to t gives


Note that if t represents time then ds/dt is speed and dR/dt is velocity. From 2) we get


From T = dR/ds we obtain



Principal normal. The principal normal, denoted by N , at a point P on a curve C, is a unit vector in the direction of dT/ds (providing dT/ds is not zero, in which case the principal normal is not defined). The principal normal is necessarily perpendicular to the unit tangent vector T. It points toward the concave side of the curve. See Fig. 4. From 4) above we obtain



Curvature of a curve at point P. The curvature κ of a curve at a point P is given by the magnitude of dT/ds:



7)        dT/ds = κN

Radius of curvature. The radius of curvature ρ is the reciprocal of the curvature:

8)        ρ = 1/κ

Center of curvature. The tip of the vector ρN, drawn from point P as initial point, is called the center of curvature of the curve at point P.


Velocity and acceleration in curvilinear motion. Consider a point P(x, y) moving along a curve defined by the parametric equations

            x = x(t)

            y = y(t)

where t is time. The position vector of the moving point is given by


The velocity is given by


The magnitude of ole20.gif is given by


where vx and vy are the x and y components of the velocity i.e.

            vx = dx/dt

            vy = dy/dt .

The direction of ole22.gif at P is along the tangent to the curve at P.

The acceleration is given by


The magnitude of ole24.gif is given by


where ax and ay are the x and y components of the acceleration i.e.

            ax = d2x/dt2

            ay = d2y/dt2 .

Tangential and normal components of acceleration. First note that





which gives the resolution of the acceleration vector at P in the directions of the unit tangent T and principal normal N. Denoting these components in the directions of T and N by ole29.gif and ole30.gif we have



where ole33.gif is the radius of curvature of the curve at P.



we have  


as a second way of determining ole36.gif .

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