document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203884>Question 1203884</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Dora was travelling a = 6km, 30° N of E, b = 3km, 60° S of E, and then c = 10km, 15° N of E. At the end of this trek, she realized that in the last leg she should have travelled c' = 4km, 75° W of N instead of the third path she originally took. In component form, what path should Dora take to reach her supposed destination? What is the displacement from her initial position? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #839921 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=839921\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i think ii have a solution.<BR>\n" );
document.write( "hopefully it's a corect solution.<BR>\n" );
document.write( "i have checked it over several times but it's easy to get twisted around so i'm keeping my fingers crossed that i did it right.\r<BR>\n" );
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document.write( "i did the calculations in excel because it's easier to keep track of them that way.<BR>\n" );
document.write( "it's still a chore, but you at least have a record that you can go back and referencde and double check without having to do all the calculations over again each time.\r<BR>\n" );
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document.write( "the component form of vercotrs is the x and y component of the triangle formed.<BR>\n" );
document.write( "when you draw a vector, the hypotenuce of the triangle formed is the distance traveled along the path of the bector.<BR>\n" );
document.write( "the horizontal leg of the right triangle is the x component and the vertical leg of the right triangle is the y component.<BR>\n" );
document.write( "the length of the x component is equal to the hypotenuse length * the cosine of the angle formed.<BR>\n" );
document.write( "the length of the y component is equal to the hypoetn7use length * the sine of the angle formed.<BR>\n" );
document.write( "the x component is assumed to extend horizontally from the vertex of the angle.<BR>\n" );
document.write( "the vetical component is assumed to extended from the end of the horizontal line to the end of the hypotenuse.<BR>\n" );
document.write( "the diagrams i created will show these relaationships.<BR>\n" );
document.write( "the base side of the triangle formed needs to be on the horizontal axis.\r<BR>\n" );
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document.write( "i drew some sketches of the angles formed.<BR>\n" );
document.write( "point O is the origin of the overall graph.<BR>\n" );
document.write( "it is assumed to be at point (0,0).\r<BR>\n" );
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document.write( "the graph sketches are shown below.<BR>\n" );
document.write( "O to A is from the origin to point A.<BR>\n" );
document.write( "A to B is from point A to point B.<BR>\n" );
document.write( "B to C is from point B to point C.<BR>\n" );
document.write( "B to C' is from point B to point C'.<BR>\n" );
document.write( "C to C' is from point C to point C'.\r<BR>\n" );
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document.write( "in order to locate the different points on the overall graph, it was necessary to calculate the horizonal and vertical distances from the origin to each point.\r<BR>\n" );
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document.write( "O to A is from the origin to point A.<BR>\n" );
document.write( "O to B is from the origin to point B.<BR>\n" );
document.write( "O to C is from the origin to point C.<BR>\n" );
document.write( "O to C' is from the origin to point C'.\r<BR>\n" );
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document.write( "the vertical individual distances from each point are combined to form the overall distance from the origin to that point.<BR>\n" );
document.write( "likewise, the horizontal distances from each point are combined to form the ocerall distance from the origin to that point.<BR>\n" );
document.write( "a case in point is from O to C.\r<BR>\n" );
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document.write( "for the overall vertical distance, you get 3 - 2.598 + 2.588 = 2.99.<BR>\n" );
document.write( "O to A is added because it is going up.<BR>\n" );
document.write( "A to B is subtracted becqause it is going down.<BR>\n" );
document.write( "B to C is added because it is going up.\r<BR>\n" );
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document.write( "for the overall horizontal distance, you get 5.196 + 1.5 + 9.659 = 16.355.<BR>\n" );
document.write( "O to A is added because it is going to the right.<BR>\n" );
document.write( "A to B is added because it is going to the right.<BR>\n" );
document.write( "B to C is added because it is going to the right.\r<BR>\n" );
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document.write( "when you go from B to C', the sketch needed to be adjusted because the base side of the triangle formed needed to be on the horizontal axis and not the vertical axis.<BR>\n" );
document.write( "doing that, preserved the convention that the x component is equal to the hypotenuse * the cosine of the angle, and the y component is equal to the hypotenuse * the sine of the angle.<BR>\n" );
document.write( "the adjustment was from 75 degrees west of east converted to 15 degrees east of west.<BR>\n" );
document.write( "that allowed the x component of the triangle formed to line up with the vertex of the angle.<BR>\n" );
document.write( "this made use of the fact that sine(angle) = cosine(90 - angle).<BR>\n" );
document.write( "75 degrees west of north is the same as 15 degrees north of west because 15 degrees + 75 degrees = 90 degrees.<BR>\n" );
document.write( "for B to C', 15 degrees north of west was modelled rather than 75 degrees west of north.\r<BR>\n" );
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document.write( "if you need further clarification of this, let me know and i'll do my best to make it more understandable.\r<BR>\n" );
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document.write( "the graph skethces are shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2023/092901.jpg\">\r<BR>\n" );
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document.write( "the calculations were performed in excel because it was easier to keep track of them tht way and reuse of calculations alread done was facilitated.<BR>\n" );
document.write( "it sill took a lot of work and a lot of thinking, but the record of what was done was easier to display.<BR>\n" );
document.write( "all numbers were rounded to 3 dcimal places for easer of viewing.<BR>\n" );
document.write( "the calculations, however, were performed with the unrounded numbers that were stored, but not displayed.\r<BR>\n" );
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document.write( "here is what the spreadsheet looked like.\r<BR>\n" );
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document.write( "since i wanted to show the angle in degrees and excel worked with angle in radians, i need to convert from degrees to radians so excel could perform the trig functions on the angle.<BR>\n" );
document.write( "that's why you see the angle in degrees and in radians.<BR>\n" );
document.write( "the formula for converting degrees to radians is radians = degrees * pi / 180.<BR>\n" );
document.write( "the formula for converting radians to degrees is degrees = radians * 180 / pi.\r<BR>\n" );
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document.write( "if  there is anything about this spreadsheet that confuses you, let me know and i'll explain as best i can.\r<BR>\n" );
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document.write( "finally, i plotted the points on a graph to show the overall paths and the order in which they were taken.<BR>\n" );
document.write( "the path lines are straiaght, but i gent them to avoid crossing out the locataion coordinates shown.<BR>\n" );
document.write( "first she went to A, then to B, then to C, then to C'.<BR>\n" );
document.write( "here's the graph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2023/092904.jpg\">\r<BR>\n" );
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document.write( "i'll be available to answer any questions you might have.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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