Question

Indicate which direction t is increasing in:
r(t)=cos(t)i-cos(t)j+sin(t)k

Answer 1

Can you elaborate what you mean by the direction it increases in?
\[
r(t) = i\cos t-j\cos t+k\sin t\\
r'(t)=-i\sin t+j\sin t+k\cos t
\]

Answer 2

I'm not sure. It just says to indicate an arrow in the direction in which t increases.

Answer 3

How would I go about graphing this equation?

Answer 4

\[\text{I'm not entirely sure, since I haven't taken a Calc II course yet, but}\\
\text{you can rearrange the function as follows...}\\
\ \ \ \hat r(t)=[\hat i-\hat j]\cos t+\hat k\sin t\\
\text{... which appears to be quite a bit like an ellipse.}\\
\ \ \ r(t)=[a\cos t]\hat i + [b\sin t]\hat j\\
\]

Answer 5

yeah I should probably know this, but how would I graph that? This is Calc. 3!

Answer 6

http://www.wolframalpha.com/input/?i=ParametricPlot3D%5B%7BCos%5Bt%5D%2C+-Cos%5Bt%5D%2C+Sin%5Bt%5D%7D%2C+%7Bt%2C+0%2C+4pi%7D%5D

Answer 7

Ok, now that that part is down. Do you know what direction t would be increasing? I believe it is asking in which direction is it expanding in

Answer 8

I believe I found the relevant parts in my Calculus book... it appears it wants an arrow to indicate the orientation, e.g. clockwise or counterclockwise.

Answer 9

Ahh! What does that mean?!

Answer 10

Essentially just pick an interval for t (e.g. 0 to 3pi/2) and then watch in what direction the curve goes as t increases. You can just change the interval in the plot on wolfram to witness it happen. It appears to be clockwise :-)

Answer 11

Thanks!

Answer 12

You seem to know how to use Wolfram really well! How would I check it on Wolfram?

Answer 13

WolframAlpha enables you use to use a subset of their Mathematica domain-specific language; one of the functions they expose is ParametricPlot3D :-p To change the interval...
ParametricPlot3D[{Cos[t],-Cos[t],Sin[t]},{t,0,pi/2}]
Look at the last parameter to the ParametricPlot3D function.
{t,0,pi/2}
t is the independent variable, and [0,pi/2] is the interval to sample. You can vary the intervals and then watch the curve grow. If you're willing to pay, you can get access to the Pro features, one of which allows you to interact with the plots.

Answer 14

Thank you! I have a hard time believing that you haven't taken Calc 3! You definitely know what you're doing!

Answer 15

Haha nah... I think there's a way to make an animated plot but I don't think WolframAlpha supports it. The full Mathematica product does, but that costs even more money...

Answer 16

By the way, verify the orientation we found... I'm not *positive* it's clockwise.

Answer 17

It is correct!