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6"a) Write the parametrized equations of the line LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ through the point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ parallel to the position vector 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.
b) Find a normal vector 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 for the for the plane LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEoJiM4NDczO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ which contains the right outside face of the parallelepiped shown in the figure.
c) Write the simplified equation for this plane.
d) Let 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 be the main diagonal of the parallepiped. Find the length LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRJHBhckYnRjJGNUYyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= of its projection along the vector 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e) Evaluate |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| and then 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[This is the scalar projection perpendicular to 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]f) Does 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 as it should?
solutiona) We introduce the position vectors and the two relevant difference 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a) 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b)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)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)PkkiYkc2IiwoSSNyMUdGJCIiIkkjcjJHRiRGJ0kjcjNHRiRGJw==PkkmcjNoYXRHNiIqJkkjcjNHRiQiIiItSSVzcXJ0R0YkNiMtSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSSxUeXBlc2V0dGluZ0c2JEYuSShfc3lzbGliR0YkNiRGJkYmISIiThe absolute value of the dot product of the unit vector with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= gives the length of its projection along that unit vectorPkklYnBhckc2Ii1JJGFic0clKnByb3RlY3RlZEc2Iy1JMGRlbGF5RG90UHJvZHVjdEc2JEYnL0krbW9kdWxlbmFtZUdGJEksVHlwZXNldHRpbmdHNiRGJ0koX3N5c2xpYkdGJDYkSSZyM2hhdEdGJEkiYkdGJA==e)QyctSTJkZWxheUNyb3NzUHJvZHVjdEc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nRzYkRiZJKF9zeXNsaWJHRik2JEkjcjNHRilJImJHRikiIiItSSVzcXJ0R0YrNiMtSTBkZWxheURvdFByb2R1Y3RHRiU2JEkiJUdGKUY3RjA+SSZicGVycEdGKSomRjdGMC1GMjYjLUY1NiRGLkYuISIiWe get the same result by crossing the unit vector into LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and taking its length of course.QyUtSTJkZWxheUNyb3NzUHJvZHVjdEc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nRzYkRiZJKF9zeXNsaWJHRik2JEkmcjNoYXRHRilJImJHRikiIiItSSVzcXJ0R0YrNiMtSTBkZWxheURvdFByb2R1Y3RHRiU2JEkiJUdGKUY3f) yes it checksQyU2JCokSSVicGFyRzYiIiIjKiRJJmJwZXJwR0YmRiciIiIvLCZGJEYqRihGKi1JMGRlbGF5RG90UHJvZHVjdEc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiZJLFR5cGVzZXR0aW5nRzYkRjBJKF9zeXNsaWJHRiY2JEkiYkdGJkY3LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=TTdSMApJNlJUQUJMRV9TQVZFLzE1MTY0MTczMlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiQiJCIiJUYnIiIkRiY=TTdSMApJNlJUQUJMRV9TQVZFLzE1MTY0MjE4MFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiQiJCMiIiMiIiQjIiIiRilGJ0YmTTdSMApJNlJUQUJMRV9TQVZFLzE1MTY0MzQ2MFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiQiJCEiJiIiIyIiJUYmTTdSMApJNlJUQUJMRV9TQVZFLzE1MTY0Mzc4MFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiQiJCMhIiYiIiQjIiIjRikjIiIlRilGJg==