document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178658>Question 1178658</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the magnitude and direction of the resultant for Vq = 5 ms and Vp 6ms. with angle 120 degrees between the two lines. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850314 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=850314\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down how to find the magnitude and direction of the resultant vector.\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Represent the Vectors:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Vq:** Magnitude = 5 ms, Direction = 0 degrees (we can assume it's along the x-axis for simplicity).<BR>\n" );
document.write( "    * Vq = 5i + 0j<BR>\n" );
document.write( "* **Vp:** Magnitude = 6 ms, Direction = 120 degrees.<BR>\n" );
document.write( "    * Vp_x = 6 * cos(120°) = 6 * (-1/2) = -3 ms<BR>\n" );
document.write( "    * Vp_y = 6 * sin(120°) = 6 * (√3 / 2) = 3√3 ms<BR>\n" );
document.write( "    * Vp = -3i + 3√3j\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Find the Resultant Vector (Vr):**\r<BR>\n" );
document.write( "\n" );
document.write( "* Vr = Vq + Vp<BR>\n" );
document.write( "* Vr = (5i + 0j) + (-3i + 3√3j)<BR>\n" );
document.write( "* Vr = (5 - 3)i + (0 + 3√3)j<BR>\n" );
document.write( "* Vr = 2i + 3√3j\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Find the Magnitude of the Resultant (||Vr||):**\r<BR>\n" );
document.write( "\n" );
document.write( "* ||Vr|| = √(Vr_x² + Vr_y²)<BR>\n" );
document.write( "* ||Vr|| = √(2² + (3√3)²)<BR>\n" );
document.write( "* ||Vr|| = √(4 + 27)<BR>\n" );
document.write( "* ||Vr|| = √31<BR>\n" );
document.write( "* ||Vr|| ≈ 5.57 ms\r<BR>\n" );
document.write( "\n" );
document.write( "**4. Find the Direction of the Resultant (θ):**\r<BR>\n" );
document.write( "\n" );
document.write( "* θ = arctan(Vr_y / Vr_x)<BR>\n" );
document.write( "* θ = arctan(3√3 / 2)<BR>\n" );
document.write( "* θ = arctan(5.196 / 2)<BR>\n" );
document.write( "* θ = arctan(2.598)<BR>\n" );
document.write( "* θ ≈ 68.96 degrees\r<BR>\n" );
document.write( "\n" );
document.write( "**Therefore:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Magnitude of the Resultant:** √31 ms (approximately 5.57 ms)<BR>\n" );
document.write( "* **Direction of the Resultant:** Approximately 68.96 degrees (measured counter-clockwise from the x-axis, or the direction of Vq).<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );