# If the lines are to intersect there must be numbers t

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If the lines are to intersect, there must be numbers t 1 and t 2 so 3 + 2t 1 = 10 + t 2 , 1 - t 1 = 8 + 3t 2 , and 2 - 2t 1 = -5 - t 2 . [Here t 1 is the parameter for the putative point in terms of the first equation and t 2 is the parameter value for the second equation.] Solving this system yields t 1 = 2 and t 2 = -3. Using either t 1 or t 2 in the appropriate vector equation yields the point of intersection, (7,-1,-2). _________________________________________________________________ 19. (5 pts.) What is the area of the triangle in three space with vertices at P = (1, 0, 0), Q = (0, 2, 0), and R = (0 , 0, 3). Let v be the vector with initial point P and terminal point Q, and let w be the vector with initial point P and terminal point R. Then the area A of the triangle is given by A = v × w /2 = <-1,2,0> × <-1,0,3> /2 = <6,3,2> /2 = 7/2. _________________________________________________________________ 20. (5 pts.) Do the three 2-space sketches of the traces in each of the coordinate planes of the surface defined by y = 1 - x 2 - z 2 . Work below and label carefully. Then attempt to do a 3 - space sketch in the plane of the surface on the back of page 4.
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