document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203355>Question 1203355</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Point a in the cartesian plane is at (2,3). Say that A is at the polar coordinates of (r, theta). If the cartesian coordinates of the point with polar coordinates (2r, theta + pi/2) are (x1, y1), and the cartesian coordinates of the point with polar coordinates (-r, -theta) are (x2,y2), what are x1,y1,x2, and y2? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838850 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=838850\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Given: A(2,3) = (r,theta)<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C200%2C-10%2C10%2C-5%2C5%2C%0D%0Aline%28-5%2C0%2C5%2C0%29%2Cline%280%2C-5%2C0%2C5%29%2C%0D%0Ared%28line%280%2C0%2C2%2C3%29%29%2C%0D%0Alocate%280.5%2C4%2C%22A%282%2C3%29=%28r%2Ctheta%29%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(1) If B is (2r,theta+pi/2) then find B(x1,y1)<br><BR>\n" );
document.write( "The angle theta+pi/2 is rotated pi/2 radians (90 degrees) counterclockwise from theta.  Calling the origin O, the vector OA has slope 3/2, so vector OB (perpendicular to OA) has slope -2/3.<BR>\n" );
document.write( "And B is twice as far from the origin as A.<BR>\n" );
document.write( "Rotated 90 degrees from A(2,3) and twice as far from the origin puts B at (-6,4).<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C200%2C-10%2C10%2C-5%2C5%2C%0D%0Aline%28-5%2C0%2C5%2C0%29%2Cline%280%2C-5%2C0%2C5%29%2C%0D%0Ared%28line%280%2C0%2C2%2C3%29%29%2C%0D%0Alocate%280.5%2C4%2C%22A%282%2C3%29=%28r%2Ctheta%29%22%29%2C%0D%0Agreen%28line%280%2C0%2C-6%2C4%29%29%2C%0D%0Alocate%28-9%2C5%2C%22B%28-6%2C4%29=%282r%2Ctheta%2Bpi%2F2%29%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(I don't know why the \"2r\" displays as \"2\r\"....)<br><BR>\n" );
document.write( "ANSWER: (x1,y1) = (-6,4)<br><BR>\n" );
document.write( "(2) If C is (-r,-theta) then find C(x2,y2)<br><BR>\n" );
document.write( "(r,-theta) is the reflection in the x-axis of A(2,3), so (r,-theta) is (2,-3).<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C200%2C-10%2C10%2C-5%2C5%2C%0D%0Aline%28-5%2C0%2C5%2C0%29%2Cline%280%2C-5%2C0%2C5%29%2C%0D%0Ared%28line%280%2C0%2C2%2C3%29%29%2C%0D%0Alocate%280.5%2C4%2C%22A%282%2C3%29=%28r%2Ctheta%29%22%29%2C%0D%0Ablue%28line%280%2C0%2C2%2C-3%29%29%2C%0D%0Alocate%280.5%2C-3%2C%22%282%2C-3%29=%28r%2C-theta%29%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(-r,-theta) is the reflection of (r,-theta) with respect to the origin, so (-r,-theta) is (-2,3).<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C200%2C-10%2C10%2C-5%2C5%2C%0D%0Aline%28-5%2C0%2C5%2C0%29%2Cline%280%2C-5%2C0%2C5%29%2C%0D%0Ared%28line%280%2C0%2C2%2C3%29%29%2C%0D%0Alocate%280.5%2C4%2C%22A%282%2C3%29=%28r%2Ctheta%29%22%29%2C%0D%0Ablue%28line%280%2C0%2C2%2C-3%29%29%2C%0D%0Alocate%280.5%2C-3%2C%22%282%2C-3%29=%28r%2C-theta%29%22%29%2C%0D%0Agreen%28line%280%2C0%2C-2%2C3%29%29%2C%0D%0Alocate%28-9%2C4%2C%22C%28-2%2C3%29=%28-r%2C-theta%29%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: (x2,y2) = (-2,3)<br><BR>\n" );
document.write( " </SPAN>\n" );
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