Find the angle between vectors r(t) and r'(t) Question?

[chrisduluk]

Hi everyone, i have a midterm exam tomorrow night (Thursday night) and i'm looking at problems in the section of the book to help myself prepare. Can anyone answer this question for me? I have attempted to answer this problem but without the answer i don't know if i'm right.

 

1. Find the angle theta between vectors r(t) and r'(t) as a function of t.

2. Sketch the graph of theta(t)

3. Find any extrema of theta(t).

4. Find any values of t which the vectors r(t) and r'(t) are orthogonal.

 

and they give vector r(t) = t^2 ii + t ij

 

And there are no typos, that's really what the question asks.

 

First off, i have NO idea what ii and ij mean, i'm assuming they just mean i and j?

 

So far i have r(t)= t^2 + t and r'(t)=2t+1 and r(t)*r'(t)=2t^3 +t

and

theta(t) = (2t^3 +t) / (sqrt(t^4+t^2) * sqrt(4t^2 +1) )

 

Can anyone confirm this is right or wrong? And basically from here i have no idea what to do to answer 2-4 of the problem. My teacher said there would be a problem just like this on my exam, so i REALLY need to see this problem done out so i know what to do.

 

Can someone do this one out for me? It's not a homework, i just need to see it done out step by step so i can follow it, along with the answer. I have until Thursday night, so PLEASE help asap, thanks!

 

PLEASE help, i'm desperate.

 

oops i meant to add arccos in the theta(t) function, but is it still right?

[ajb]

So you have taken the dot (aka scalar) product and then used [math]a.b = |a||b| \cos \theta[/math], for vectors [math]a,b[/math]. Then as these vectors are parametrised by by [math]t[/math], you solved for [math]\theta(t)[/math]. All seems correct so far...

 

For the second part just plot the function [math]\theta(t)[/math]. You can just throw in a few vales of [math]t[/math] and see what you get.

 

For 3, you need to consider when [math]\theta'(t)= 0[/math] If there are any values of [math]t[/math] that satisfy this then we have a local stationary point at [math]t[/math].

 

Orthogonal means the [math]a.b=0[/math]. You have the dot product as a function of [math]t[/math], so are there any values of [math] t[/math] where the dot product vanishes?