Vector Questions (1 Viewer)

[kev-kun]

Bit stuck on these two questions. The first one I can do just don't know how to find the values of arg.

1) Find the modulus and argument of each of the complex numbers z1=2i and z2=1+sqrt(3)i. Mark on an Argand diagram the points P, Q, R and S representing, z1, z2, z1+z2, and z1-z2 respectively. Deduce the exact values of arg(z1+z2) and arg (z1-z2)

2) On an Argand diagram, the points A, B, C and D represend z1, z2, z3 and z4, respectively. Show that if z1-z2+z3-z4=0, then ABCD is a parallelogram, and if also z1+iz2-z3-iz4=0, then ABCD is a square.

Help much appreciated ^^

 

[integral95]

For Q 2

After marking the points in the diagram

You rearrange the equation and you get => z1-z4 = z2-z3

Therefore the vectors AD = BC

Rearranging again you get z1-z2 = z4-z3

There fore AB = CD

Since the opposite sides of the quadrilateral are equal, it's a parallelogram

For the 2nd part

Rearrange the equation z1- z3 = i(z4-z2)

Vector AC is vector BD rotated 90 degrees anti- clockwise

There fore the diagonals AC and BD of the quadrilateral are equal and intersect at right angles

There fore ABCD is a square

(sorry if it's a bit unclear)

 

[kev-kun]

No worries because it does make sense! Thanks~