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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: lisafaustina on October 31, 2010, 11:37:02 am

Title: Rays
Post by: lisafaustina on October 31, 2010, 11:37:02 am: Ok so when u have somethig like
Sketch {z: Arg(z) < -pi/4 } how do you sketch this in the complex plane? I dunno which area to shade in..
Title: Re: Rays
Post by: ghadz7 on October 31, 2010, 11:40:37 am: The angle from (0,0) is lower than -pi/4.
Title: Re: Rays
Post by: lisafaustina on October 31, 2010, 11:41:54 am: Could someone please roughly sketch this graph?
Title: Re: Rays
Post by: jto1292 on October 31, 2010, 11:42:40 am: Shade all of quadrant 3 and the half of quad 4 that's UNDER the *ray*. Have the *ray* dotted.. I think that's right. Correct me if wrong anyone..

Lol soz kept referring to the ray as arc**
Title: Re: Rays
Post by: 98.40_for_sure on October 31, 2010, 11:42:46 am: Don't forget to open circle the origin and dotted lines!
Title: Re: Rays
Post by: jto1292 on October 31, 2010, 11:47:05 am: Negative Re(z) axis should also be dotted.. and yes, open circle on origin
Title: Re: Rays
Post by: JinXi on October 31, 2010, 11:48:02 am: 1 thing i feel weird about, is when they give u an axis and ask u to sketch like Arg(z)=-pi/4, how are we supposed to dot the -ve x axis so that its clear-.-
Title: Re: Rays
Post by: 98.40_for_sure on October 31, 2010, 11:48:20 am: Lol it's hard to draw a frickin dotted line OVER their lines :( usually i use a red pen to do it but scared i can't rub it out if it turns out to be incorrect
Title: Re: Rays
Post by: jto1292 on October 31, 2010, 11:51:30 am: agreed. I always get scared I'm doing it wrong.. Thanks for this question though, coz I remembered I need to revise region shading in complex plane....
Title: Re: Rays
Post by: lisafaustina on October 31, 2010, 11:53:04 am: Thanks. What if it's like {z: Arg(z) < or = -pi/4}
and what if it's {z: Arg(z) > -pi/4}

when do you have an open circle on origin and when do you have a closed circle?
Title: Re: Rays
Post by: 98.40_for_sure on October 31, 2010, 11:55:05 am: {z: Arg(z) < or = -pi/4} is dotted -ve Re(z) axis and full line at -pi/4

{z: Arg(z) > -pi/4} is dotted line at pi/4 and full line at -ve Re(z) axis as -pi<Arg(z)< and = pi
Title: Re: Rays
Post by: jto1292 on October 31, 2010, 11:57:06 am: Rays always have open circle on their starting point. If it's Arg(z), (0,0) has open circle. If it's Arg(z-(1+3i), (1,3) has an open circle. Arg(z) < or = to -pi/4 means make the ray solid, not dotted, but KEEP negative Re(z) axis dotted. EDIT: Also keep same shading as Arg(z) < -pi/4..

Arg(z) > -pi/4 means dotted ray and shade quad 1 2 and the half of 4 that's above the ray, NO quad 4. Also dot the neg Re(z) axis.
Title: Re: Rays
Post by: JinXi on October 31, 2010, 12:02:58 pm: Quote from: jto1292 on October 31, 2010, 11:57:06 am
Rays always have open circle on their starting point. If it's Arg(z), (0,0) has open circle. If it's Arg(z-(1+3i), (1,3) has an open circle. Arg(z) < or = to -pi/4 means make the ray solid, not dotted, but KEEP negative Re(z) axis dotted. EDIT: Also keep same shading as Arg(z) < -pi/4..

Arg(z) > -pi/4 means dotted ray and shade quad 1 2 and the half of 4 that's above the ray, NO quad 4. Also dot the neg Re(z) axis.

Arg(z) > something , doesn't that mean -ve x axis is solid, since -pi < Arg(z) </= pi.
Title: Re: Rays
Post by: jto1292 on October 31, 2010, 12:05:08 pm: Quote from: JinXi on October 31, 2010, 12:02:58 pm
Quote from: jto1292 on October 31, 2010, 11:57:06 am
Rays always have open circle on their starting point. If it's Arg(z), (0,0) has open circle. If it's Arg(z-(1+3i), (1,3) has an open circle. Arg(z) < or = to -pi/4 means make the ray solid, not dotted, but KEEP negative Re(z) axis dotted. EDIT: Also keep same shading as Arg(z) < -pi/4..

Arg(z) > -pi/4 means dotted ray and shade quad 1 2 and the half of 4 that's above the ray, NO quad 4. Also dot the neg Re(z) axis.

Arg(z) > something , doesn't that mean -ve x axis is solid, since -pi (greater) Arg(z) (greater of equal to) pi.

Good point. You're probs right... Neg Re(z) axis being dotted should only happen on instances such as Arg(z) > pi/4. Good pick up. Sorry about confusion :S
Title: Re: Rays
Post by: jto1292 on October 31, 2010, 12:11:51 pm: OKEY so for further clarification, Arg(z) > and = or > angle means DOT -ve Re(z). Arg(z)< or = or < angle means -ve Re(z) should be included...
Title: Re: Rays
Post by: JinXi on October 31, 2010, 12:15:21 pm: Quote from: jto1292 on October 31, 2010, 12:11:51 pm
OKEY so for further clarification, Arg(z) > and = or > angle means DOT -ve Re(z). Arg(z)< or = or < angle means -ve Re(z) should be included...

Thats upside down. What 99.95 4 sure posted is right Arg> include -ve Re(z), Arg < exclude. Thats it for x-axis.
Title: Re: Rays
Post by: jto1292 on October 31, 2010, 12:20:26 pm: But doesn't Arg(z) have a range of -pi < angle < or = pi ?? In saying that, wouldn't that mean that if Arg(z) > angle it will INCLUDE the negative Re(z) axis because it reads it as +pi ? and if so, wouldn't Arg(z) < / = angle mean that -pi ( -ve Re(z) ) is not included thus dotted because of the range?
Title: Re: Rays
Post by: JinXi on October 31, 2010, 12:22:41 pm: what i meant by -pi < Arg(z) </= pi is that for all values of Arg(z) must lie between -pi(excluded) to pi (included)
Title: Re: Rays
Post by: jto1292 on October 31, 2010, 12:24:16 pm: Good, yes, sounds right (Y)
Title: Re: Rays
Post by: lisafaustina on October 31, 2010, 12:31:17 pm: ok is this correct?
i just did this on paint
Title: Re: Rays
Post by: jasoN- on October 31, 2010, 12:32:15 pm: if that is including pi, then yes
Title: Re: Rays
Post by: JinXi on October 31, 2010, 12:33:39 pm: yeah the -ve x-axis should be solid.
Title: Re: Rays
Post by: lisafaustina on October 31, 2010, 12:37:40 pm: aw what i thought -ve x-axis was always dotted :SSS
Title: Re: Rays
Post by: jto1292 on October 31, 2010, 03:25:00 pm: IF the angle's positive, -ve x-axis is solid, if the angle is -ve, the -ve x-axis is dotted.
Title: Re: Rays
Post by: jasoN- on October 31, 2010, 03:27:37 pm: not true; eg $Arg(z) \geq -\frac{\pi}{3}$
What you're trying to say is if $Arg(z) \geq \theta$ or $Arg(z) > \theta$ then -ve x-axis is solid (ie. pi)
Likewise if $Arg(z) \leq \theta$ or $Arg(z) < \theta$ then -ve x-axis is dotted
Title: Re: Rays
Post by: jto1292 on October 31, 2010, 03:39:44 pm: I'm never gonna get explaining it right, am I? :/ Oh well...
Title: Re: Rays
Post by: Elnino_Gerrard on October 31, 2010, 07:06:28 pm: Whats the difference in the diagrams of Arg(z)>=-3pie/4 and z:-3pie/4<=Arg(z)<=0 ???
Title: Re: Rays
Post by: jto1292 on October 31, 2010, 07:54:47 pm: Arg(z)>=-3pie/4 covers 1st 2nd and 4th quad fully. -ve x-axis is solid, ray is solid and half of 3rd quad that's under the ray is covered.

z:-3pie/4<=Arg(z)<=0 covers only half of third quad that's under the ray, and all of quad 4. +ve x-axis is included..
Title: Re: Rays
Post by: Elnino_Gerrard on October 31, 2010, 08:10:09 pm: ahhhhh i see cheers mate.