Question 1194519: 1. Given: Z1= 2-2i , Z2= 3i and Z3= -3+i . Solve analytically and illustrate graphically. (Illustrate all 3 vectors and the result vector.)
(a) Z1-Z2+Z3 (b) Z3-Z2-Z1
Answer by proyaop(69)   (Show Source):
You can put this solution on YOUR website! **1. Analytical Solutions**
**(a) Z1 - Z2 + Z3**
* Z1 - Z2 + Z3 = (2 - 2i) - 3i + (-3 + i)
* Z1 - Z2 + Z3 = 2 - 2i - 3i - 3 + i
* Z1 - Z2 + Z3 = -1 - 4i
**(b) Z3 - Z2 - Z1**
* Z3 - Z2 - Z1 = (-3 + i) - 3i - (2 - 2i)
* Z3 - Z2 - Z1 = -3 + i - 3i - 2 + 2i
* Z3 - Z2 - Z1 = -5
**2. Graphical Illustration**
* **Represent complex numbers as vectors:**
* Z1 = 2 - 2i: Vector from origin to point (2, -2) in the complex plane.
* Z2 = 3i: Vector from origin to point (0, 3) in the complex plane.
* Z3 = -3 + i: Vector from origin to point (-3, 1) in the complex plane.
* **Perform vector operations graphically:**
* **(a) Z1 - Z2 + Z3:**
* Draw Z1.
* Draw -Z2 (vector Z2 in the opposite direction).
* Draw Z3.
* The vector sum Z1 - Z2 + Z3 is the resultant vector obtained by connecting the tail of Z1 to the head of Z3 after drawing -Z2.
* **(b) Z3 - Z2 - Z1:**
* Draw Z3.
* Draw -Z2.
* Draw -Z1 (vector Z1 in the opposite direction).
* The vector sum Z3 - Z2 - Z1 is the resultant vector obtained by connecting the tail of Z3 to the head of -Z1 after drawing -Z2.
**3. Further Calculations**
**(c) Z1 * Z3**
* Z1 * Z3 = (2 - 2i) * (-3 + i)
* Z1 * Z3 = -6 + 2i + 6i - 2i²
* Z1 * Z3 = -6 + 8i + 2 (since i² = -1)
* Z1 * Z3 = -4 + 8i
**(d) Z3 x Z2**
* The cross product is not defined for complex numbers in the same way it is for vectors in 3D space.
**(e) Acute Angle between Z1 and Z2**
* Find the magnitudes of Z1 and Z2:
* |Z1| = √(2² + (-2)²) = √8 = 2√2
* |Z2| = √(0² + 3²) = 3
* Find the dot product of Z1 and Z2:
* Z1 • Z2 = (2 * 0) + (-2 * 3) = -6
* Use the dot product formula:
* cos(θ) = (Z1 • Z2) / (|Z1| * |Z2|)
* cos(θ) = -6 / (2√2 * 3)
* cos(θ) = -√2 / 2
* θ = 135°
* The acute angle between Z1 and Z2 is 180° - 135° = 45°.
**Note:**
* Graphical representation can be done using a complex plane (Argand diagram).
* You can use graphing software or tools to plot the complex numbers and visualize the vector operations.
I hope this comprehensive explanation helps!
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