complex number question

Question

in an argand diagram the points a and b represent 1+7i and 4+3i respectively and o is the origin the quadrilateral oabc is a parallelogra find the complex numbler represented by point c

Answer 1

Assuming oabc follows standard nomenclarure, i.e, the verticies are in the same order as the letters, then oc is parallel to ab.

Draw and label a diagram.

You shall see that c is 3 - 4i.

Answer 2

c is 5+10i

c=1+7i+4+3i= 1+4+7i+3i= 5+10i

but that's a parallelogram better described as 0acb putting the 4 points in that order

the problems does say 0abc, so another parallelgram is formed by

c as 3-4i

there are two possible parallelograms formed with the points a, b, c and the origin

graph the points with imaginary numbers measured on the vertical, y -axis and real numbers on the horizontal or x-axis

then vertex c is between vertices b and 0 a=4+3i b=1+7i

real component of c is 4-1 = 3

the imaginary compoent is 3i-7i = -4i

c is then 3-4i

c = a-b = 4+3i-1-7i = 3-4i

the slope from c to a = the slope from the origin to b, for the lines to be parallel

from a to b has slope = (7-3)/(1-4) = -4/3 = y/x, -4x= 3y, 4x+3y = 0 where c = x+yi

slope 0 to a has slope = (7-0)/(1-0) = 7 = (3-y)/(4-x)

28-7x = 3-y

7x-y = 25

4x+3y=0

21x-3y=75

add

25x = 75

x = 75/25 = 3

-4/3=y/3

y =-4

x+yi = 3-4i = c

c= 3 -4i

thre's a 3rd possibility for c which creates a parallelogram

c=-3 +4i

the three possibilities can be described in vertex form in order

as 0abc, 0cab and 0acb

or as 0cba, 0bac and 0bca

each with a different value for c

for the quadrant I c, c=a+b both real and imaginary components >0 5+10i

for the quadrant IV c, c=b-a imaginary component <0, real >0 3-4i

for quadrant II c, c= a-b imaginary component >0, real <0 -3+4i

you want quadrant IV

c=b-a = 4+3i - (1+7i) = 4+3i -1-7i = 3-4i