Geometric Representation Question (1 Viewer)

[Nashchnc]

Having a bit of difficulty with a question and unsure whether my approach is correct?

"z1=4-i, z2=2i and z3 form vertices of an isosceles right angled triangle, with z3 the right angle, Find z3."

Is the question, my working went along the lines of:

Vector z3,z1 = (i) x vector z3,z2.

(z1-z3)=(i)(z2-z3)
4 - i - z3 = -2 - (i)z3
4 - i +2 = z3(1-i)
(6 - i)/(1 - i) = z3

Multiplying by conjugate gives

(7+5i)/2



Can someone kindly point out my flaw? Ive drawn it up on an argand plane and it doesnt seem like 90 degrees at that point? Thanls in advance

 

[anomalousdecay]

Multiplying the numerator and denominator gives us:



I got the same thing as you...... I drew the diagram as well. Did you copy the question out properly?

 

[obliviousninja]

Can't there be two points for Z3? The vertice could be on the top right or bottom left?

 

[obliviousninja]

Having a bit of difficulty with a question and unsure whether my approach is correct?

"z1=4-i, z2=2i and z3 form vertices of an isosceles right angled triangle, with z3 the right angle, Find z3."

Is the question, my working went along the lines of:

Vector z3,z1 = (i) x vector z3,z2.

(z1-z3)=(i)(z2-z3)
4 - i - z3 = -2 - (i)z3
4 - i +2 = z3(1-i)
(6 - i)/(1 - i) = z3

Multiplying by conjugate gives

(7+5i)/2



Can someone kindly point out my flaw? Ive drawn it up on an argand plane and it doesnt seem like 90 degrees at that point? Thanls in advance

Do you have answers?

 

[Nashchnc]

no i dont have answers, its from the pittwater house paper on here

 

[QZP]

@OP, you are correct. There is no mistake...

 

[QZP]

There is another answer though

 

[Nashchnc]

I thought so, thanks guys

As a side note, the locus arg((z-i)/(z+1))=pi/2

Will be a semicirle between i and -1, where a line between the two is the diameter of the locus, which is a circle?

 

[Nashchnc]

How do i get the other result?

EDIT: I think its multiplication by -i am i correct???

 

[QZP]

How do i get the other result?

EDIT: I think its multiplication by -i am i correct???

It's the exact same thing as what you did for the first answer. Just z3 is in a different location... (hint: consider where z3 is if it is reflected about the axis joining z1 and z2)

I thought so, thanks guys

As a side note, the locus arg((z-i)/(z+1))=pi/2

Will be a semicirle between i and -1, where a line between the two is the diameter of the locus, which is a circle?

What's your answer? Semi circle or a circle? I can't help you if I don't know where you're going wrong :S

 

[Nashchnc]

answer is a semicircle in quadrant 2, starting at i, ending at -1?

 

[QZP]

answer is a semicircle in quadrant 2, starting at i, ending at -1?

It's not exactly a "semi-circle" (diameter not included as part of locus), and there is also a restriction on z which you haven't mentioned...

 

[obliviousninja]

I fairly sure you also have to take into account the other scenario for Z3

ie. if the point was in the bottom left, rather than top right. (z3-z1)i = z3-z2

 

[HeroicPandas]

I fairly sure you also have to take into account the other scenario for Z3

ie. if the point was in the bottom left, rather than top right. (z3-z1)i = z3-z2

I agree with this 2013%. There's nothing wrong with this idea

 

[obliviousninja]

I agree with this 2013%. There's nothing wrong with this idea

Yes! I have approval from Panda!

 

[anomalousdecay]

Yes! I have approval from Panda!

You have my approval as well.