MAT2500-03/04 10S 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), draw in the three points and their position vectors, labeling each point and vector, and draw in the triangle the points determine, labeling the three sides by appropriate symbols for their difference vectors (put in arrow heads to indicate which direction your vector symbol for each side refers to).
b) Evaluate the angle of the triangle at the vertex P1 exactly (no decimals) and numerically to the nearest tenth of a degree. Does your result seem consistent with your graph? Explain.
c) On the reverse side of this sheet (right grid), draw in only the position vectors of the first two points and draw in the rectangle used to graphically project the second position vector parallel and perpendicular to the first vector, labeling the vectors and sides of the rectangle with appropriate notation for those projections. Estimate roughly the 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 your numerically evaluated exact vectors compare to your graphical estimates? Explain.solutiona) We introduce the position vectors.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Qyg+SSNyMUc2Ii1JJDwsPkdJKF9zeXNsaWJHRiU2JCIiIiIiIyEiIj5JI3IyR0YlLUYnNiRGLCIiJUYsPkkjcjNHRiUtRic2JCIiJiIiJEYsHere is the blank for the quiz sheet.LUkoZGlzcGxheUc2IjYjLUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYoIiIhOyEiIyIiJi9JJmNvbG9yR0YkSSZibGFja0dGJC9JKHNjYWxpbmdHRiRJLGNvbnN0cmFpbmVkR0YkL0kqZ3JpZGxpbmVzR0YkSSV0cnVlR0YpL0kldmlld0dGJDckRi1GLQ==QzMtSSZhcnJvd0c2IjYlSSNyMUdGJS9JJnNoYXBlR0YlRiQvSSZjb2xvckdGJUkkcmVkR0YlISIiPkkjcDFHRiVJIiVHRiVGLS1GJDYlSSNyMkdGJUYoL0YrSSVibHVlR0YlRi0+SSNwMkdGJUYwRi0tRiQ2JUkjcjNHRiVGKC9GK0kmZ3JlZW5HRiVGLT5JI3AzR0YlRjBGLS1JJXBsb3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JDcjLUkkbWFwR0ZCNiVJKGNvbnZlcnRHRkI3JkYnRjNGOkYnSSVsaXN0R0ZCL0YrSSZibGFja0dGJUYtPkkjcDRHRiVGMEYtLUkoZGlzcGxheUdGJTYpRi9GN0Y+Rk8vSShzY2FsaW5nR0YlSSxjb25zdHJhaW5lZEdGJS9JKmdyaWRsaW5lc0dGJUkldHJ1ZUdGQi9JJXZpZXdHRiU3JDshIiMiIiZGZm4=But this just shows the triangle; we need to add in the difference vectors as requested to represent the three sides.Qyc+SSRyMjFHNiIsJkkjcjJHRiUiIiJJI3IxR0YlISIiRig+SSRyMzFHRiUsJkkjcjNHRiVGKEYpRipGKD5JJHIzMkdGJSwmRi5GKEYnRio=Here we put in the difference vectors.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 pretty but it was not worth all the effort to create the 6 symbols on the diagram!
b) Next we find the corresponding lengths and unit vectors needed for the angle at the first vertexJSFHQyk+SSdyMjFtYWdHNiItSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMtSTBkZWxheURvdFByb2R1Y3RHNiRGKS9JK21vZHVsZW5hbWVHRiVJLFR5cGVzZXR0aW5nR0YoNiRJJHIyMUdGJUYzIiIiLUkmZXZhbGZHRik2JEkiJUdGJSIiJUY0PkkncjMxbWFnR0YlLUYnNiMtRi02JEkkcjMxR0YlRkBGNEY1QyY+SSdyMjFoYXRHNiIqJkkkcjIxR0YlIiIiLUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLUkwZGVsYXlEb3RQcm9kdWN0RzYkRiwvSSttb2R1bGVuYW1lR0YlSSxUeXBlc2V0dGluZ0dGKzYkRidGJyEiIkYoPkkncjMxaGF0R0YlKiZJJHIzMUdGJUYoLUYqNiMtRjA2JEY6RjpGNkYoThen we compute their dot product and arccosine and convert to degreesQyktSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiRJJ3IyMWhhdEdGKUkncjMxaGF0R0YpIiIiLUknYXJjY29zR0YrNiNJIiVHRilGMC1JJmV2YWxmR0YmNiMsJComRjRGMEkjUGlHRiYhIiIiJCE9RjAqJi1GNjYkRjQiIiVGMEkoZGVncmVlc0dGKUYwd)Here is the blank for the quiz sheet.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JSFHc) 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.PkkmcjFoYXRHNiIqJkkjcjFHRiQiIiItSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMtSTBkZWxheURvdFByb2R1Y3RHNiRGKy9JK21vZHVsZW5hbWVHRiRJLFR5cGVzZXR0aW5nR0YqNiRGJkYmISIiQzEtSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0c2JEYmSShfc3lzbGliR0YpNiRJI3IyR0YpSSZyMWhhdEdGKSIiIi1JJmV2YWxmR0YmNiRJIiVHRikiIiVGMD5JJnIycGFyR0YpLUYkNiVJIyUlR0YpRi9JJXRydWVHRiZGMC1GMjYjRjRGMD5JJ3IycGVycEdGKSwmRi5GMEY6ISIiRjBGPEYwL0YuLCZGN0YwRj9GMEYwLUYkNiRGN0Y/Then some maple trick to get a list of points for plotting the rectangle.NyMtSSRtYXBHJSpwcm90ZWN0ZWRHNiVJKGNvbnZlcnRHJSpwcm90ZWN0ZWRHNyZJI3IxRzYiSSZyMXBhckc2IkkjcjJHNiJJJ3IycGVycEc2IkklbGlzdEclKnByb3RlY3RlZEc=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I am out of patience for the symbol labels. The parallel vector component is the green arrow, the cyan arrow is the perpendicular vector component of the original blue vector along the red vector (whose arrow head is the only part showing).