document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210495>Question 1210495</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Calculate the area of the triangle formed by the complex numbers , , and  in the Argand plane, where  is a complex number. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852934 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852934\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The area of the triangle formed by the complex numbers $z_1 = 0$, $z_2 = 1$, and $z_3 = i$ in the Argand plane is **$\frac{1}{2}$ square units**.\r<BR>\n" );
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document.write( "***\r<BR>\n" );
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document.write( "### Step 1: Identify the Coordinates\r<BR>\n" );
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document.write( "In the Argand plane, a complex number $z = x + yi$ corresponds to the Cartesian coordinate point $(x, y)$.\r<BR>\n" );
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document.write( "1.  $z_1 = 0 = 0 + 0i \implies P_1 = (0, 0)$<BR>\n" );
document.write( "2.  $z_2 = 1 = 1 + 0i \implies P_2 = (1, 0)$<BR>\n" );
document.write( "3.  $z_3 = i = 0 + 1i \implies P_3 = (0, 1)$\r<BR>\n" );
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document.write( "### Step 2: Determine the Type of Triangle\r<BR>\n" );
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document.write( "Plotting these points reveals a right-angled triangle:\r<BR>\n" );
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document.write( "* The side from $P_1$ to $P_2$ lies along the positive **Real axis** (the $x$-axis) and has a length of **1**.<BR>\n" );
document.write( "* The side from $P_1$ to $P_3$ lies along the positive **Imaginary axis** (the $y$-axis) and has a length of **1**.<BR>\n" );
document.write( "* Since the Real and Imaginary axes are perpendicular, the angle at the origin ($P_1$) is $90^\circ$.\r<BR>\n" );
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document.write( "### Step 3: Calculate the Area\r<BR>\n" );
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document.write( "The area of a right-angled triangle is given by the formula:\r<BR>\n" );
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document.write( "$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$\r<BR>\n" );
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document.write( "In this case:<BR>\n" );
document.write( "* Base $= \text{length of the side from } (0, 0) \text{ to } (1, 0) = 1$<BR>\n" );
document.write( "* Height $= \text{length of the side from } (0, 0) \text{ to } (0, 1) = 1$\r<BR>\n" );
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document.write( "$$\text{Area} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}$$\r<BR>\n" );
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document.write( "***\r<BR>\n" );
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document.write( "### Alternative Method: Using the Determinant Formula\r<BR>\n" );
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document.write( "The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is also given by:\r<BR>\n" );
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document.write( "$$\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$$\r<BR>\n" );
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document.write( "Plugging in the coordinates $(0, 0)$, $(1, 0)$, and $(0, 1)$:\r<BR>\n" );
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document.write( "$$\text{Area} = \frac{1}{2} \left| 0(0 - 1) + 1(1 - 0) + 0(0 - 0) \right|$$<BR>\n" );
document.write( "$$\text{Area} = \frac{1}{2} \left| 0 + 1(1) + 0 \right|$$<BR>\n" );
document.write( "$$\text{Area} = \frac{1}{2} \times 1 = \frac{1}{2}$$ </SPAN>\n" );
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