Fundamentals of Engineering: Complex Numbers P3

Let z_1 = 2 — i and z_2 = 1 + i. Use the parallelogram law to construct each of the following vectors:

• z_1 + z_2

• z_1 — z_2

• 2z_1–3z_2

SOLUTION

To solve the problem, we treat each complex number as a vector in the complex plane and perform the required operations.

. a) Construct z_1 + z_2

z_1 + z_2 = (2 — i) + (1 + i)

= 2 + 1 + (-i + i)

= 3 + 0i

= 3

This result represents the vector 3, corresponding to the point (3, 0) on the complex plane.

. b) Construct z_1 — z_2

z_1 — z_2 = (2 — i) — (1 + i)

= 2–1 + (-i — i)

= 1–2i

This result represents the vector 1–2i, corresponding to the point (1, -2) on the complex plane.

. c) Construct 2z_1–3z_2

2z_1–3z_2 = 2(2 — i) — 3(1 + i)

= (4–2i) — (3 + 3i)

= 4–3 + (-2i — 3i)

= 1–5i

This result represents the vector 1–5i, corresponding to the point (1, -5) on the complex plane.

SUMMARY OF RESULTS

• z_1 + z_2 = 3

• z_1 — z_2 = 1–2i

• 2z_1–3z_2 = 1–5i