Vector Functions

[d0minicz]

A circle of radius 5 has its centre at the point C with position vector 2i + 6j relative to the origin O. A general point P on the circle has position r relative to O. The angle between i and measured in anticlockwise sense from i to is denoted by .
a) Give the vector equation of P

thanks !..

[TrueTears]

In general the parametric equation for a circle are and

P is the locus of points on the circle with centre (2,6).

Hence and

since it denotes the radius.

Therefore : and

[d0minicz]

A particle travels in a path such that the position vector r(t) at time t is given by r(t) = 3cos(t)i + 2sin(t)j ,.
a) Express this vector function as a cartesian relation
b) Find the initial position of the particle
c) If the positive y axis points north and the positive x axis points east, find correct to two decimal places, the bearing of the point P, the position of the particle at from:
i) the origin
ii) the initial position

thanks !!

[TrueTears]

a) let x = 3cos(t) and y = 2sin(t)

and



b) initial occurs when at r(0)

c) i. Find the vector then use

[d0minicz]

oh thanks whered you get the last formula from ?

[TrueTears]

oh thanks whered you get the last formula from ?

It's not a formula, it's called drawing a graph.

[d0minicz]

rofl alright another

An object is moving so that its position r at time t is given by r(t) =
a) Find the initial position of the object
b) Find the position at
c) Find the cartesian equation of the path.

thx again

[kamil9876]

http://vcenotes.com/forum/index.php/topic,15142.0.html

[TrueTears]

WARNING THE FOLLOWING IS NOT IN THE SPESH COURSE:

let and

now and

[d0minicz]

Hey how do we know if a particle is moving anti-clockwise or clockwise?

[TrueTears]

Try a value of t and see what coordinate that is on the cartesian.

EDIT: 3000 posts.

[kamil9876]

several ways of working this out. Assuming we already know it is moving circularly, it suffices to find three points, the point where , and the point when (It's analogous to finding where the particle will be at midnight. 6am and noon) where P is the period of the function. Sketch these points and jump from one to the other in a chronological way and see what kind of motion (clockwise or anticlockwise) would be neccesary to jump in such an order.

[TrueTears]

several ways of working this out. Assuming we already know it is moving circularly, it suffices to find three points, the point where , and the point when (It's analogous to finding where the particle will be at midnight. 6am and noon) where P is the period of the function. Sketch these points and jump from one to the other in a chronological way and see what kind of motion (clockwise or anticlockwise) would be neccesary to jump in such an order.

If you have calculator, just pick whatever t that seems reasonable.

[kamil9876]

True, just make sure that it's within the same cycle/period. That's why I picked those 3 points. Example of a mistake is say when the period is 10 but you use the points at t=3,t=12,t=21. You get the opposite direction in that case. (draw and see)

edit: better example provided

[/0]

If the cartesian equation is a hyperbola then what exactly do you mean by clockwise and anticlockwise motion?

As for ellipses, say you have an equation of the form:

,





Now, if and are the same sign, then the direction is anticlockwise.
If and are of opposite sign, then the direction is clockwise.

Beware: If you have then the situation is reversed. (Make sure you see why this is)