a2 solu - MATH 251-3 Fall 2007 Simon Fraser University...

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MATH 251-3, Fall 2007 Simon Fraser University Assignment 2: Solutions Additional Question Due: 4:30pm, Monday 24 September 2007 1. Consider the four given points A (4 , - 2 , 3), B (5 , 0 , - 1), C (4 , - 3 , 2) and D (2 , - 7 , 10). (a) Find an equation for the line through the points A and B . Solution: [Note: These solutions give many details of the steps of the calculations; you do not need to show all of these steps.] A direction vector for the line is u = -→ AB = -→ OB - -→ OA = 1 , 2 , - 4 ; and the point A (4 , - 2 , 3) , with position vector r 0 = -→ OA = 4 , - 2 , 3 is a point on the line. Hence the vector equation of the line is (where r = x, y, z is the position vector of an arbitrary point on the line) r = r 0 + t u = 4 , - 2 , 3 + t 1 , 2 , - 4 = 4 + t, - 2 + 2 t, 3 - 4 t , where t R is a parameter. Alternative forms: The parametric equations for the line are thus x = 4 + t, y = - 2 + 2 t, z = 3 - 4 t, and the symmetric equations (non-parametric) are (solving for t ) x - 4 1 = y + 2 2 = z - 3 - 4 . Note: Since I didn’t specify a particular form of the equation for the line, you were free to give any one (or more!) of the above forms.
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