This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 2263 Fall 2009 Midterm 1, WITH SOLUTIONS October 8, 2009 1. (15 points) The lines given parametrically by ( x, y, z ) = (1 + t, 2 2 t, 2 t ) ,∞ < t < ∞ and ( x, y, z ) = (1 + 2 s, 2 + 2 s, s ) ,∞ < s < ∞ intersect at the point ( x, y, z ) = (1 , 2 , 0) . Find an equation for the plane which contains both lines . Write the equation in the form ax + by + cz = d. SOLUTION: A normal vector ~v is the cross product of the vectors ~ i 2 ~ j +2 ~ k and 2 ~ i +2 ~ j + ~ k , which are vectors in the directions of the two given lines. ~v = ~ i ~ j ~ k 1 2 2 2 2 1 = 6 ~ i + 3 ~ j + 6 ~ k. We can divide ~v by 3, so an equation for the plane is 2( x 1)+( y +2)+2 z = 0, or simplifying: 2 x + y + 2 z = 4 . 2. (13 points) Find an equation for the hyperbolic cylinder in ( x, y, z )space containing in finitely many lines parallel to the xaxis, and containing the slanted hyperbola x = y, x 2 + y 2 z 2 = 4 . SOLUTION: Eliminate x from the equations x = y , x 2 + y 2 z 2 = 4 to get 2 y 2 z 2 = 4 ....
View
Full Document
 Spring '08
 Staff
 Math, Derivative, Multivariable Calculus, 2J, 2k, 4k, 6k, 3j

Click to edit the document details