MAT2500-01/04 12S Quiz 2 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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 and the parallelopiped formed from their three position vectors 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. [Note 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 form the front face of this object, with 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 extending backwards from this face.]
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Write the parametrized equations of the line LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ through the top right edge of the parallelopiped as shown. [What is the simplest position vector of a point on the line? What is the orientation of the line?]
b) Find a normal vector 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 for the plane LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEoJiM4NDczO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRJHRvcEYnRjJGNUYyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= which contains the top face of the parallelepiped shown in the figure.
c) Write the simplified equation for this plane.
d) Find the scalar projection LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= of 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 along 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.
e) Evaluate the area LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= of the bottom face of the parallelogram, a parallelogram formed by the edges 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f) Does the volume V=|LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RInxnckYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZMLUY2Ni1RIn5GJ0Y5RjtGPi9GQUY9RkJGREZGRkgvRktRJjAuMGVtRicvRk5GVC1GLDYlUSJBRidGL0YyRjk= of the parallelopiped equal the triple scalar product |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 as it should?
solutiona) We introduce the position vectors and the two relevant difference 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) The front top right corner of the top has the position vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjcjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIrRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkwtRiw2JVEjcjNGJ0YvRjJGOQ==, while the orientation is along LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcjJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn: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) e) the magnitude of this cross product is the area of the parallelogram face formed by these two 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