## Wednesday, February 27, 2013

### Complex Numbers Solutions

The Problems again:

1) Find the sum of (1 –i) and (6 – 6i)

Solution:

(1 - i) + (6 - 6i) = (1 + 6) + (-i - 6i) = 7 - 7i

2) Find the sum of (0- i) and (15 – 0i)

Solution:

(0 - i) + (15 - 0i) = (0 + 15) + (-i - 0) = 15 -i

3) Subtract: (3 + 3i) from 10 – 4i

Solution:

(10 - 4i)  - (3 + 3i) =  (10 - 3) + (-4i - (3i)) = 7 - 7i

4) Find: (1 –i) + (11 – 2i) – (4 – 4i)

Solution:  Write:

[(1 - i) + (11 - 2i)] - (4 - 4i)

Then: [(1 - i) + (11 - 2i)] = 12 - 3i

Suibtracting:

(12 - 3i) - (4 - 4i) = (12 - 4) + (-3i) - (-4i) = 8 + i

5) Represent the two parts of (1) as vectors on an Argand diagram, and show their resultant.

Solution: The vector construction procedure will be analogous to that depicted in Fig. 1 of the Complex numbers blog, except in this case the vectors are added end to end, starting with the first (ending at 1,-1) then the next, culminating with the resultant vector terminating at (7, -7) in the lower right quadrant of the Argand diagram. The overall resultant is at 45 degrees with respect to each axis (x, y), which is the same orientation as each of the component vectors. (How do you confirm this?)

6) Using the Argand diagram of Fig. 1 and complex addition, find the resultant of the vector sum B + C.

B = (2 - 3i) and C = (4 + 3i)

So vector addition will yield a resultant for a parallelogram (you need to complete the other two sides, e.g. B' and C') extending from the point (2,-3) and ending at the point (4,3) which is 6 units in length. That is, conforming to:

(2 + 4) + (-3i + 3i) =  6