Parametric curves and complex numbers

• Sep 2nd 2011, 10:57 PM
andrew2322
Parametric curves and complex numbers
Hello guys, i've done a few more problems that i need corrected, thanks.

1.) The position of a particle in the xy plane at time t (seconds) is given by r(t) = (2 + 5t^2)i + (-3 + t)j

a) find the cartesian equation of the particles trajectory

my answer: x = 5y^2 + 30y +47

b) Sketch the particles trajectory also indicating its direction of motion

my solution: the shape of the graph x = 5y^2 + 30y + 47 is a sideways parabola with a vertex at (2,-3) which was found by completing the square. the particle is travelling in the direction to increase y.

__________________________________________________ ______________________

2.) Write the following complex numbers in cartesian form x + iy where x and y are real numbers.

a) (2 - 3i) / (2i - 1)

my solution: -i/8 - 8/3

b) (3-2i)^2 (1+5i)

my solution: 75 + 13i

__________________________________________________ ______________________

3.) Find the modulus and argument of z = - (root3)/(10) - 1/10i between -pi and pi.

my solution: arg(z) = - 5pi/6
modulus (z) = 1/5

__________________________________________________ _________________________
• Sep 3rd 2011, 02:51 AM
Siron
Re: Parametric curves and complex numbers
Ex 2:
a) Almost correct, you made a mistake in the denominator, it has to be:
$\frac{2-3i}{2i-1}=\frac{(2-3i)(2i+1)}{(2i-1)(2i+1)}=\frac{4i-6i^2+2-3i}{-4-1}=\frac{i+8}{-5}=\frac{-i}{5}-\frac{8}{5}$

b) Also here you made a little mistake:
$(3-2i)^2\cdot (1+5i)=(9-12i-4)\cdot (1+5i)=(5-12i)\cdot (1+5i)=5+25i-12i-60i^2=5+13i+60=65+13i$

Ex 3:
Correct!
You can do the check by yourself, if $\mbox{arg}(z)=\frac{-5\pi}{6}$ and $\mbox{mod}(z)=\frac{1}{5}$ then the complex number is:
$\frac{1}{5}\cdot \left[\cos\left(\frac{-5\pi}{6}\right)+i\sin\left(\frac{-5\pi}{6}\right)\right]=$
$\frac{-\sqrt{3}}{10}-\frac{i}{10}=z$
• Sep 3rd 2011, 03:00 AM
FernandoRevilla
Re: Parametric curves and complex numbers
Quote:

Originally Posted by andrew2322
1.) The position of a particle in the xy plane at time t (seconds) is given by r(t) = (2 + 5t^2)i + (-3 + t)j

a) find the cartesian equation of the particles trajectory my answer: x = 5y^2 + 30y +47

b) Sketch the particles trajectory also indicating its direction of motion my solution: the shape of the graph x = 5y^2 + 30y + 47 is a sideways parabola with a vertex at (2,-3) which was found by completing the square. the particle is travelling in the direction to increase y.

Right.
• Sep 3rd 2011, 03:07 AM
FernandoRevilla
Re: Parametric curves and complex numbers
Quote:

Originally Posted by andrew2322
3.) Find the modulus and argument of z = - (root3)/(10) - 1/10i between -pi and pi.
my solution: arg(z) = - 5pi/6 modulus (z) = 1/5

Right.
• Sep 3rd 2011, 06:43 PM
andrew2322
Re: Parametric curves and complex numbers
thanks guys, much appreciated.
• Sep 3rd 2011, 08:02 PM
Prove It
Re: Parametric curves and complex numbers
Quote:

Originally Posted by andrew2322
Hello guys, i've done a few more problems that i need corrected, thanks.

1.) The position of a particle in the xy plane at time t (seconds) is given by r(t) = (2 + 5t^2)i + (-3 + t)j

a) find the cartesian equation of the particles trajectory

my answer: x = 5y^2 + 30y +47

b) Sketch the particles trajectory also indicating its direction of motion

my solution: the shape of the graph x = 5y^2 + 30y + 47 is a sideways parabola with a vertex at (2,-3) which was found by completing the square. the particle is travelling in the direction to increase y.

__________________________________________________ ______________________

2.) Write the following complex numbers in cartesian form x + iy where x and y are real numbers.

a) (2 - 3i) / (2i - 1)

my solution: -i/8 - 8/3

b) (3-2i)^2 (1+5i)

my solution: 75 + 13i

__________________________________________________ ______________________

3.) Find the modulus and argument of z = - (root3)/(10) - 1/10i between -pi and pi.

my solution: arg(z) = - 5pi/6
modulus (z) = 1/5

__________________________________________________ _________________________
\displaystyle \begin{align*} 5y^2 + 30y + 47 &= x \\ y^2 + 6y + \frac{47}{5} &= \frac{x}{5} \\ y^2 + 6y + 3^2 + \frac{47}{5} &= \frac{x}{5} + 3^2 \\ \left(y + 3\right)^2 + \frac{47}{5} &= \frac{x + 45}{5} \\ \left(y + 3\right)^2 &= \frac{x - 2}{5} \\ \left(y + 3\right)^2 &= \frac{5(x - 2)}{25} \\ \left(y + 3\right)^2 &= \frac{5x - 10}{25} \\ y + 3 &= \pm \frac{\sqrt{5x - 10}}{5} \\ y &= \frac{-15 \pm \sqrt{5x-10}}{5}\end{align*}
I can't guess why we should write $y$ in terms of $x$ .