Complex Number Question (1 Viewer)

[let.me.die]

hi.. i just started complex numbers and i was wondering if someone could help me with some questions.
The points P and Q are represented by the complex numbers z = 1-3i and w = -3+4i respectively. FInd a point R on the real azis such that PRQ is a right-angled triangle.

 

[Porcia]

no bloody idea, actually i'd like to find out too... seems like we're doing same question

 

[Raginsheep]

Umm...let z be represented by P(1,-3) and w by Q(-3,4). Call R (x,0).

Use gradient formula to find gradient of PR and QR. Gradient PR x Gradient QR = -1. Then, solve for x.

 

[let.me.die]

thanx. i was wondering if i could just use co-ordinate geometry to find the answer. i got other questions if someone wouldnt mind answering them..
1. If |z| = |w|, prove that (z+w)/(z-w) is purely imaginary. By drawing a suitable diagram, give a geometrical interpretation of the result.

2. Prove that for any two complex numbers z1 and z2 |z1 + z2| more than or equal to |z1| - |z2|, assuming |z1| > |z2|. When does the equality sign hold?

 

[Raginsheep]

1. Multiply top and bottom by the conjugate of (z-w). The bottom becomes |z-w|^2 which is real.

For the top, its (z+w)(z-w). Assume the underline means conjugate.
(z+w)(z-w)
=(z+w)(z-w)
=zz-ww-zw+zw
=|z|^2-|w|^2-zw+zw
But, |z|=|w|
=zw-zw
let z=(x1, y1), w=(x2, y2) and sub in.
Now expand and simplify and the real terms should disappear leaving only the imaginary.

Since top is imaginay and bottom is real, the whole thing is imaginary.

 

[robbo_145]

thanx. i was wondering if i could just use co-ordinate geometry to find the answer. i got other questions if someone wouldnt mind answering them..
1. If |z| = |w|, prove that (z+w)/(z-w) is purely imaginary. By drawing a suitable diagram, give a geometrical interpretation of the result.

drawing a diagram it can be seen that z+w and z-w are the diagonals of a rhombus of sides, hence they cross at right angles
i assume you've done De Moivre's thereom

let z+w = rCisΘ
z-w= qCis(Θ-π/2) //q,r are real
(z+w)/(z-w)= r/qCis(Θ-(Θ-π/2)) by De Moivre
(z+w)/(z-w) = r/qCis(π/2) which is purely imaginary