document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191444>Question 1191444</A>: <I><SPAN STYLE=\"word-break: break-all;\"> B. Magnitude of Vectors (Trigonometry)<BR>\n" );
document.write( "3. Find the magnitude and direction of the sum w of the following vectors. Express the direction as a measure of the angle between v and w.<BR>\n" );
document.write( "- If ||u|| = 6.5, ||v|| = 7.2 and the angle between them measures 150 degrees.\r<BR>\n" );
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document.write( "Show full solution, thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849099 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.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=849099\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the magnitude and direction of the sum of the vectors u and v:\r<BR>\n" );
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document.write( "**1. Find the x and y components of each vector:**\r<BR>\n" );
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document.write( "* **Vector u:**<BR>\n" );
document.write( "    * u_x = ||u|| * cos(θ_u) = 6.5 * cos(θ_u)<BR>\n" );
document.write( "    * u_y = ||u|| * sin(θ_u) = 6.5 * sin(θ_u)\r<BR>\n" );
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document.write( "* **Vector v:**<BR>\n" );
document.write( "    * v_x = ||v|| * cos(θ_v) = 7.2 * cos(θ_v)<BR>\n" );
document.write( "    * v_y = ||v|| * sin(θ_v) = 7.2 * sin(θ_v)\r<BR>\n" );
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document.write( "We need to establish a reference angle. Let's assume vector *u*'s angle θ_u is 0 degrees. Then vector *v*'s angle θ_v is 150 degrees.\r<BR>\n" );
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document.write( "* **Vector u:**<BR>\n" );
document.write( "    * u_x = 6.5 * cos(0°) = 6.5<BR>\n" );
document.write( "    * u_y = 6.5 * sin(0°) = 0\r<BR>\n" );
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document.write( "* **Vector v:**<BR>\n" );
document.write( "    * v_x = 7.2 * cos(150°) = 7.2 * (-√3/2) ≈ -6.24<BR>\n" );
document.write( "    * v_y = 7.2 * sin(150°) = 7.2 * (1/2) = 3.6\r<BR>\n" );
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document.write( "**2. Find the x and y components of the resultant vector w:**\r<BR>\n" );
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document.write( "* w_x = u_x + v_x = 6.5 + (-6.24) ≈ 0.26<BR>\n" );
document.write( "* w_y = u_y + v_y = 0 + 3.6 = 3.6\r<BR>\n" );
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document.write( "**3. Find the magnitude of w:**\r<BR>\n" );
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document.write( "* ||w|| = sqrt(w_x² + w_y²) = sqrt(0.26² + 3.6²) ≈ sqrt(0.0676 + 12.96) ≈ sqrt(13.0276) ≈ 3.61\r<BR>\n" );
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document.write( "**4. Find the direction of w:**\r<BR>\n" );
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document.write( "* θ_w = arctan(w_y / w_x) = arctan(3.6 / 0.26) ≈ arctan(13.85) ≈ 85.9°\r<BR>\n" );
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document.write( "Since w_x is positive and w_y is positive, the angle is in the first quadrant, so 85.9° is correct.\r<BR>\n" );
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document.write( "**5. Find the angle between u and w:**\r<BR>\n" );
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document.write( "The angle between u and w is simply θ_w - θ_u = 85.9° - 0° = 85.9°.\r<BR>\n" );
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document.write( "**Answers:**\r<BR>\n" );
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document.write( "* **Magnitude of w:** Approximately 3.61<BR>\n" );
document.write( "* **Direction of w (relative to u):** Approximately 85.9°<BR>\n" );
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