document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194517>Question 1194517</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. Given: Z1= 2-2i , Z2= 3i and Z3= -3+i . Solve analytically and illustrate graphically. (Illustrate all 3 vectors and the result vector.)<BR>\n" );
document.write( " (a) Z1-Z2+Z3 (b) Z3-Z2-Z1\r<BR>\n" );
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document.write( "2. Using the same given as Problem 1. Find:<BR>\n" );
document.write( " (c) Z1*Z3   (d) Z3xZ2   (e) the acute angle between Z1 and Z2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848459 by </B><A HREF=/tutors/aboutme.mpl?userid=proyaop><B>proyaop(69)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=proyaop><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848459\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Analytical Solutions**\r<BR>\n" );
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document.write( "**(a) Z1 - Z2 + Z3**\r<BR>\n" );
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document.write( "* Z1 - Z2 + Z3 = (2 - 2i) - 3i + (-3 + i) <BR>\n" );
document.write( "* Z1 - Z2 + Z3 = 2 - 2i - 3i - 3 + i <BR>\n" );
document.write( "* Z1 - Z2 + Z3 = -1 - 4i\r<BR>\n" );
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document.write( "**(b) Z3 - Z2 - Z1**\r<BR>\n" );
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document.write( "* Z3 - Z2 - Z1 = (-3 + i) - 3i - (2 - 2i) <BR>\n" );
document.write( "* Z3 - Z2 - Z1 = -3 + i - 3i - 2 + 2i <BR>\n" );
document.write( "* Z3 - Z2 - Z1 = -5 \r<BR>\n" );
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document.write( "**2. Graphical Illustration**\r<BR>\n" );
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document.write( "* **Represent complex numbers as vectors:**<BR>\n" );
document.write( "    * Z1 = 2 - 2i: Vector from origin to point (2, -2) in the complex plane.<BR>\n" );
document.write( "    * Z2 = 3i: Vector from origin to point (0, 3) in the complex plane.<BR>\n" );
document.write( "    * Z3 = -3 + i: Vector from origin to point (-3, 1) in the complex plane.\r<BR>\n" );
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document.write( "* **Perform vector operations graphically:**<BR>\n" );
document.write( "    * **(a) Z1 - Z2 + Z3:** <BR>\n" );
document.write( "        * Draw Z1. <BR>\n" );
document.write( "        * Draw -Z2 (vector Z2 in the opposite direction). <BR>\n" );
document.write( "        * Draw Z3.<BR>\n" );
document.write( "        * 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. \r<BR>\n" );
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document.write( "    * **(b) Z3 - Z2 - Z1:**<BR>\n" );
document.write( "        * Draw Z3.<BR>\n" );
document.write( "        * Draw -Z2.<BR>\n" );
document.write( "        * Draw -Z1 (vector Z1 in the opposite direction).<BR>\n" );
document.write( "        * 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.\r<BR>\n" );
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document.write( "**3. Further Calculations**\r<BR>\n" );
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document.write( "**(c) Z1 * Z3**\r<BR>\n" );
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document.write( "* Z1 * Z3 = (2 - 2i) * (-3 + i) <BR>\n" );
document.write( "* Z1 * Z3 = -6 + 2i + 6i - 2i² <BR>\n" );
document.write( "* Z1 * Z3 = -6 + 8i + 2 (since i² = -1)<BR>\n" );
document.write( "* Z1 * Z3 = -4 + 8i\r<BR>\n" );
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document.write( "**(d) Z3 x Z2**\r<BR>\n" );
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document.write( "* The cross product is not defined for complex numbers in the same way it is for vectors in 3D space. \r<BR>\n" );
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document.write( "**(e) Acute Angle between Z1 and Z2**\r<BR>\n" );
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document.write( "* Find the magnitudes of Z1 and Z2:<BR>\n" );
document.write( "    * |Z1| = √(2² + (-2)²) = √8 = 2√2<BR>\n" );
document.write( "    * |Z2| = √(0² + 3²) = 3\r<BR>\n" );
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document.write( "* Find the dot product of Z1 and Z2:<BR>\n" );
document.write( "    * Z1 • Z2 = (2 * 0) + (-2 * 3) = -6\r<BR>\n" );
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document.write( "* Use the dot product formula:<BR>\n" );
document.write( "    * cos(θ) = (Z1 • Z2) / (|Z1| * |Z2|)<BR>\n" );
document.write( "    * cos(θ) = -6 / (2√2 * 3) <BR>\n" );
document.write( "    * cos(θ) = -√2 / 2<BR>\n" );
document.write( "    * θ = 135° \r<BR>\n" );
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document.write( "* The acute angle between Z1 and Z2 is 180° - 135° = 45°.\r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* Graphical representation can be done using a complex plane (Argand diagram). <BR>\n" );
document.write( "* You can use graphing software or tools to plot the complex numbers and visualize the vector operations.\r<BR>\n" );
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document.write( "I hope this comprehensive explanation helps!<BR>\n" );
document.write( " </SPAN>\n" );
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