MAT2500-01/02 20S Quiz 4 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 EQUAL SIGNS and arrows 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). Only use technology to CHECK hand calculations, not substitute for them.Given the vector-valued function 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 for the domain LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEvJkdyZWF0ZXJFcXVhbDtGJ0YyL0Y2USdub3JtYWxGJy8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk4tSSNtbkdGJDYlUSIwRidGMkY8RjJGPA== (no credit for unidentified expressions):
a) Evaluate 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 and remember to simplify your results.
b) Evaluate 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, 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 and remember to simplify your results.
c) Evaluate the exact angle \316\270 in radians between 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 and 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 and a single decimal place approximation in degrees. Does it seem compatible with the figure shown on the reverse side of this sheet? Why?
d) Evaluate the vector 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 which is the vector projection of 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 orthogonal (perpendicular!) to 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.
e) Optional. Use the diagram on the reverse side to include a diagram of this projection.solutiona) We introduce the position vectors and the two relevant difference 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a)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b)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRj5GPi1JI21vR0YkNi1RIjtGJ0Y+LyUmZmVuY2VHRkIvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGKy1GRDYtUSInRidGPkZHL0ZKRkJGS0ZNRk9GUUZTL0ZWUSwwLjExMTExMTFlbUYnL0ZZRldGNUZDRitGZW5GZW5GNUZARj4=JSFHa) again and b)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c)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d)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This tells us that the paralle projection along the tangent vector is about 1.2 times its length, enabling us to nail down where the perpendicular from the tip of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtSSNtbkdGJDYlUSIxRidGMi9GNlEnbm9ybWFsRidGMkZBRjJGQUYyRkE= with initial point at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtSSNtbkdGJDYlUSIxRidGMi9GNlEnbm9ybWFsRidGMkZBRjJGQUYyRkE= falls along the tangent vector.e) Here is the 3d plot: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We scale down the normal a bit so it does not distort the picture: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Now some play to plot the various projections and projection 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Of course we could have also drawn the vector resolution diagram at the origin as well.Here is an animation of the curve with the triad of unit vectors (set the number of frames to 11):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JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYzLUkjbWlHRiQ2JlEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtSSNtbkdGJDYlUSIxRidGMi9GNlEnbm9ybWFsRidGMkZBRjJGQS1JI21vR0YkNi5RIixGJ0YyRkEvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJ0YrLUZENi5RIidGJ0YyRkFGRy9GSkY0RktGTUZPRlFGUy9GVlEsMC4xMTExMTExZW1GJy9GWUZXRjhGQy1GRDYuUSJ+RidGMkZBRkdGaG5GS0ZNRk9GUUZTRlVGW29GK0ZlbkZlbkY4RkMtSSVtc3ViR0YkNiYtRjk2JS1GIzYmRitGOEYyRkFGMkZBLUYjNiYtRkQ2LlEnJnBlcnA7RidGMkZBRkdGaG5GS0ZNRk9GUUZTL0ZWUSwwLjI3Nzc3NzhlbUYnL0ZZRlxwRi9GMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1GRDYuUSIuRidGMkZBRkdGaG5GS0ZNRk9GUUZTRlVGW29GMkZB (only the first two matter here, perp with respect to 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)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TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ1NzI2NzA5MTI5NDg2WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQjIiIjIiIkIyIiIkYrRilGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ1NzI2NzA5MTMwMDMwWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQsJCokIiIjIyIiIkYrI0YrIiIkLCRGKiNGLSIiJ0YwRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ1NzI2NzA5MTMxMzkwWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQjIiNBIiIqIyIjNkYrRilGJQ==TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ1NzI2NzA5MTMyMDcwWColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQjIiM5IiIqIyEiI0YrIyEjOEYrRiU=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ1NzI2NzA5MTI1MTE4WColKWFueXRoaW5nRzYiLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIkIiQjIiIiIiIjIyEiJEYrI0YqIiIlRiU=