MAT2500-03/04 14S Quiz 1 Print Name (Last, First) _________________________________|__Show all work, including mental steps, in a clearly organized way that speaks for itself. Use proper mathematical notation, identifying expressions by their proper symbols (introducing them if necessary), and use arrows and equal signs when appropriate. Always simplify expressions. BOX final short answers. LABEL parts of problem. Keep answers EXACT (but give decimal approximations for interpretation). Indicate where technology is used and what type (Maple, GC).Given three points 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 in the plane:a) On the reverse side of this sheet (left grid), BEFORE doing this problem, draw in the three points and their position vectors, labeling each point and vector (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 or 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 etc), and draw in the triangle that the points determine, labeling the two sides 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUYjNictSSNtaUdGJDYjUSFGJy1GIzYkLUkmbW92ZXJHRiQ2JS1GIzYmLUYsNiUtRjE2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0ZEUSdub3JtYWxGJ0ZARkMvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLDYlRj0tRiM2JS1GSTYkUSIzRidGTEZARkNGTkZARkMtSSNtb0dGJDYuUS0mUmlnaHRBcnJvdztGJy8lLHBsYWNlaG9sZGVyR0ZCRkwvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRmpuLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRmpuLyUobGFyZ2VvcEdGam4vJS5tb3ZhYmxlbGltaXRzR0Zqbi8lJ2FjY2VudEdGam4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0Zpby9GZm9GQkZMRjBGQEZDLUYjNiVGMEZARkNGTkYwRkw= by appropriate symbols for their difference vectors and put in arrow heads to indicate which direction your vector symbol for each side refers to.
b) First give a very rough estimate of the angle of the triangle at the vertex LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ , then evaluate it exactly (no decimals) and numerically to the nearest tenth of a degree (don't change your initial estimate!). Does your result seem consistent with estimate? Explain. [Not even I can estimate an angle better than 10 degrees without a protractor, but it should be in the same ballpark, so to speak.]
c) On the reverse side of this sheet, draw in the rectangle used to graphically project 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 parallel and perpendicular to 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 (use a straight edge of piece of paper to draw the lines), and draw in the vectors 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 and 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 labeling the vectors and sides of the rectangle with appropriate notation for those projections. Estimate roughly the numerical components of the two vector projections.
d) Now using appropriate notation, step by step (show every step starting from the initial vector components), evaluate the vector components of the parallel and perpendicular projections that you have drawn exactly and then to 2 decimal place accuracy.
e) How do the numerical evaluations of your exact vectors compare to your graphical estimates? Do they seem consistent? Explain.solutiona) We introduce the position vectors.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Here is the blank for the quiz 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But this just shows the triangle; we need to add in the difference vectors as requested to represent the three sides.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 we put in the two difference vectors from the first point.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b) Next we find the corresponding lengths and unit vectors needed for the angle at the first vertexLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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Then we compute their dot product and arccosine and convert to degrees. My guesstimate is that is like 120 degrees, but who can really guess that well an angle?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)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=c) Now we evaluate the scalar projection of the first vector along the second, and then multiply the direction of the second by this scalar, then subtract from the first vector to get the orthogonal vector component. Then evaluate the length of the orthogonal vector component.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 almost like a square but not quite (seen next plot). The sides have lengths: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Then some maple trick to get a list of points for plotting the rectangle. We start with the first position vector and then add successive tip to tail vectors as we move around the rectangle counterclockwise to get the 4 corners, but we have to convert the list of vectors into a list of pairs of coordinates to plot, which is what the map, convert stuff does.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 I were forced to guesstimate the componets of the vectors along the two sides of this rectangle, I would say the perpendicular vector is about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtbkdGJDYkUSQxLjlGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYkUSQxLjdGJ0Y0RjRGNC8lJW9wZW5HUScmbGFuZztGJy8lJmNsb3NlR1EnJnJhbmc7RidGNA== and the parallel vector about 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. Compare: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 bad.Now we add it into our plot.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Nice.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kiISIiLSUpQk9VTkRTX1lHNiMkIiQ/IiEiIi0lLUJPVU5EU19XSURUSEc2IyQiJV1QISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJV1QISIiLSUpQ0hJTERSRU5HNiI=