MAT2500-01/02 20S Quiz 3 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 subsitute for them.
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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 comes forward to the front face of this object, with the parallelogram 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 equaling the back face.]
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Write the parametrized equations of the line LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RkE= through the right edge of the front face LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEpJiM1ODUzOTtGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiZRKmZyb250ZmFjZUYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNS9GOVEnbm9ybWFsRic= as shown. [What is the simplest position vector of a point on the line? What is the orientation of the line?] Where does this line intersect the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8 plane?
b) Find a normal vector 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 for the plane LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEpJiM1ODUzOTtGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiZRKmZyb250ZmFjZUYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNS9GOVEnbm9ybWFsRic= which contains the front face of the parallelepiped shown in the figure.
c) Write the simplified equation for this plane. Does the point at the tip of the main diagonal of the parallelopiped (from the origin to the opposite corner) satisfy this equation as it should?
d) Find the scalar projection LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn of the main diagonal of the parallelopiped along 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. [LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LlEifkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRS1JKG1mZW5jZWRHRiQ2Jy1GIzYmLUkjbWlHRiQ2JlEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnRi8vRjNRJ2l0YWxpY0YnRitGL0YyRi9GMi8lJW9wZW5HUSJ8Z3JGJy8lJmNsb3NlR0ZYRi9GMg== is just the distance of the front face plane from the origin, or its height if we instead think of that face as the top of the parallelopiped.]
e) Evaluate the area LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn of the front face of the parallelopiped, a parallelogram formed by the edges parallel to 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.f) Does the volume V=|LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEifGdyRidGMi9GNlEnbm9ybWFsRicvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZOLUY5Ni5RIn5GJ0YyRjxGPkZAL0ZDRjRGREZGRkhGSi9GTVEmMC4wZW1GJy9GUEZWLUYsNiZRIkFGJ0YvRjJGNUYyRjw= 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 vectorsLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlMtRiw2JVEld2l0aEYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRKFN0dWRlbnRGJ0YvRjItRlk2Ji1GIzYlLUYsNiVRLkxpbmVhckFsZ2VicmFGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Zhb0Y5RjlGNUZPRlUtRlk2JC1GIzYlLUYsNiVRKnBsb3R0b29sc0YnRi9GMkZhb0Y5RjlGNUZVLUZZNiQtRiM2JC1GLDYlUSZwbG90c0YnRi9GMkY5RjlGNUZhb0Y5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYyLUkjbWlHRiQ2JVEjcjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JKG1mZW5jZWRHRiQ2Ji1GIzYqLUkjbW5HRiQ2JFEiM0YnRjktRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJy1GVTYkUSIxRidGOUZYLUY2Ni1RKiZ1bWludXMwO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZhb0Zqbi8lK2V4ZWN1dGFibGVHRj1GOUY5LyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJy1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0Znbi8lJmRlcHRoR0ZjcC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GLDYmUSNyMkYnRi9GY29GMi1GNjYuRjhGY29GOUY7Rj5GQEZCRkRGRkZIRkpGTS1GUDYnLUYjNipGXW9Gam5GWC1GVTYkUSIyRidGOUZYRmpuRmNvRjlGY29GOUZlb0Zoby1GNjYuRl1wRmNvRjlGO0Y+RkBGQkZERkZGSEZKRk1GXnAtRiw2JlEjcjNGJ0YvRmNvRjJGXnEtRlA2Jy1GIzYpRmpuRlhGam5GWEZkcUZjb0Y5RmNvRjlGZW9GaG9GZ3FGY29GOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVEoZGlzcGxheUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYwLUYsNiVRJmFycm93RidGL0YyLUY2NiQtRiM2MS1GLDYlUSNyMUYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGTC8lKnN5bW1ldHJpY0dGTC8lKGxhcmdlb3BHRkwvJS5tb3ZhYmxlbGltaXRzR0ZMLyUnYWNjZW50R0ZMLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEmc2hhcGVGJ0YvRjItRkU2LVEiPUYnRkhGSi9GTkZMRk9GUUZTRlVGVy9GWlEsMC4yNzc3Nzc4ZW1GJy9GZ25GYW9GOkZELUYsNiVRJmNvbG9yRidGL0YyRlxvLUYsNiVRJHJlZEYnRi9GMkZELUYsNiVRKnRoaWNrbmVzc0YnRi9GMkZcby1JI21uR0YkNiRRIjJGJ0ZILyUrZXhlY3V0YWJsZUdGTEZIRkhGRC1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0Zlbi8lJmRlcHRoR0ZncC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJ0Y6LUY2NiQtRiM2MC1GLDYlUSNyMkYnRi9GMkZERmluRlxvRjpGREZjb0Zcby1GLDYlUSVibHVlRidGL0YyRkRGaW9GXG8tRl1wNiRRIjNGJ0ZIRkhGSC1GRTYtUSJ+RidGSEZKRl9vRk9GUUZTRlVGV0ZZL0ZnbkZlbkZERmJwRjotRjY2JC1GIzYwLUYsNiVRI3IzRidGL0YyRkRGaW5GXG9GOkZERmNvRlxvLUYsNiVRJmdyZWVuRidGL0YyRkRGaW9GXG9GXHBGSEZIRlxyRmBwRkhGSC1GRTYtUSI6RidGSEZKRl9vRk9GUUZTRlVGV0Zgb0Zib0Zcci1GLDYlUSpwbHRhcnJvd3NGJ0YvRjItRkU2LVEqJmNvbG9uZXE7RidGSEZKRl9vRk9GUUZTRlVGV0Zgb0Ziby1GLDYlUSIlRidGL0YyRmpyRmBwRkg=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUkjbWlHRiQ2JVEoZGlzcGxheUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYyLUYsNiVRL3BhcmFsbGVsZXBpcGVkRidGL0YyLUY2NiQtRiM2KS1GLDYlUSNyMUYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGTC8lKnN5bW1ldHJpY0dGTC8lKGxhcmdlb3BHRkwvJS5tb3ZhYmxlbGltaXRzR0ZMLyUnYWNjZW50R0ZMLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEjcjJGJ0YvRjJGRC1GLDYlUSNyM0YnRi9GMi8lK2V4ZWN1dGFibGVHRkxGSEZIRkQtRiw2JVEsb3JpZW50YXRpb25GJ0YvRjItRkU2LVEiPUYnRkhGSi9GTkZMRk9GUUZTRlVGVy9GWlEsMC4yNzc3Nzc4ZW1GJy9GZ25GaW8tRjY2Ji1GIzYqLUkjbW5HRiQ2JFEiMEYnRkhGRC1GYHA2JFEjODVGJ0ZIRkQtRkU2LVEqJnVtaW51czA7RidGSEZKRmdvRk9GUUZTRlVGVy9GWlEsMC4yMjIyMjIyZW1GJy9GZ25GanAtRmBwNiRRIzMyRidGSEZfb0ZIRkgvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGRC1GLDYlUSVheGVzRidGL0YyRmRvLUYsNiVGSUYvRjJGRC1GLDYlUS10cmFuc3BhcmVuY3lGJ0YvRjJGZG8tRmBwNiRRJDAuN0YnRkhGX29GSEZILUZFNi1RIjpGJ0ZIRkpGZ29GT0ZRRlNGVUZXRmhvRmpvLUZFNi1RIn5GJ0ZIRkpGZ29GT0ZRRlNGVUZXRlkvRmduRmVuRistRjY2JC1GIzYrLUYsNiVRIiVGJ0YvRjJGRC1GLDYlUSpwbHRhcnJvd3NGJ0YvRjJGRC1GLDYlUShzY2FsaW5nRidGL0YyRmRvLUYsNiVRLGNvbnN0cmFpbmVkRidGL0YyRl9vRkhGSC1GRTYtUSI7RidGSEZKRk1GT0ZRRlNGVUZXRllGam9GY3ItRiw2JVElcHBsdEYnRi9GMi1GRTYtUSomY29sb25lcTtGJ0ZIRkpGZ29GT0ZRRlNGVUZXRmhvRmpvRltzRmByRl9vRkg=a) The rear top left corner of the bottom has the position vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEjcjJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIitGJ0YyL0Y2USdub3JtYWxGJy8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRk4tRiw2JlEjcjNGJ0YvRjJGNUYyRjw=, while the orientation is along LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEjcjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YyL0Y2USdub3JtYWxGJw==: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Where does this line hit the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8 plane?LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY5LUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LUY2Ni1RIjtGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRidGTS1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRlcvRk5GWC1GLDYlUSJURidGL0YyLUY2Ni1RKiZjb2xvbmVxO0YnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRiw2JVEmc29sdmVGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2Ky1GUDYkUSIyRidGOUZZLUYsNiVRInRGJ0YvRjJGNUZPLUY2Ni1RIixGJ0Y5RjtGVkZARkJGREZGRkhGVy9GTlEsMC4zMzMzMzMzZW1GJ0Zoby8lK2V4ZWN1dGFibGVHRj1GOUY5RlNGWS1GLDYlUSJ4RidGL0YyRjUtRiw2JVElc3Vic0YnRi9GMi1GYW82JC1GIzYrRmhvRjVGZ25GW3BGZW8tRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZgcUZob0ZgcEY5RjlGUy1GLDYlUSJ5RidGL0YyRjVGZXAtRmFvNiQtRiM2K0Zob0Y1RmduRltwLUZQNiRRIjNGJ0Y5RlxxRllGaG9GOUY5RlNGYHBGOQ==At the point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNiktSSNtbkdGJDYlUSIyRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRidGNEY3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JVEiM0YnRjRGN0Y6LUYxNiVRIjBGJ0Y0RjdGNEY3RjRGN0Y0Rjc=.
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b) e) the magnitude of this cross product is the area of the parallelogram face formed by these two 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c) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEjcjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YyL0Y2USdub3JtYWxGJw== is a position vector on this plane, 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c) Does the main diagonal satisfy this equation?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.d)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f)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRictRiw2JVEiaEYnRi9GMkY1LUY2Ni1RIn5GJ0Y5RjsvRj9GPUZARkJGREZGRkhGSi9GTkZMLUYsNiVRIlZGJ0YvRjItRjY2LVEqJmNvbG9uZXE7RidGOUY7RlZGQEZCRkRGRkZIL0ZLRk9GTUYrRlNGUC8lK2V4ZWN1dGFibGVHRj1GOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVEjcjNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLUYsNiVRI3IxRidGL0YyLUY2Ni1RKCZ0aW1lcztGJ0Y5RjsvRj9GPUZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GWC1GLDYlUSNyMkYnRi9GMkY1RistRjY2LVEpJm1pZGRvdDtGJ0Y5RjtGVkZARkJGREZGRkgvRktRLDAuMTY2NjY2N2VtRicvRk5GW28tSShtZmVuY2VkR0YkNiQtRiM2J0ZQRlNGWi8lK2V4ZWN1dGFibGVHRj1GOUY5RmJvRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=They 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