2009 Exam1 Solution

2009 Exam1 Solution - Math 2263 Fall 2009 Midterm 1 WITH...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the 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 x-axis, 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

{[ snackBarMessage ]}

Page1 / 3

2009 Exam1 Solution - Math 2263 Fall 2009 Midterm 1 WITH...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online