quiz2soln

quiz2soln - x , y , and z . Namely, we get the vector h 1 ,...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
MATH 231 / Spring 2008 January 30, 2008 Name: Key 1. (4 points) Complete the following statements with reference to material from last week’s lecture. (a) The vectors a and b are parallel if and only if a × b = 0 (b) The vectors a and b are perpendicular if and only if a · b = 0 2. (6 points) Find parametric equations for the line through the point (0 , 1 , 2) that is parallel to the plane x + y + z = 2 and perpendicular to the line x = 1 + t , y = 1 - t , z = 2 t . Solution: We wish to somehow find the equation of a line. We know that, in order to find a line, we need two things: a point on the line and a direction vector for the line. We are given the point (0 , 1 , 2), so all we have left to find is the equation of our direction vector. Since our line is parallel to the plane x + y + z = 2, we know that it is perpendicular to the plane’s normal vector. To find the normal vector, we may simply look at the coefficients in front of
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: x , y , and z . Namely, we get the vector h 1 , 1 , 1 i . Furthermore, we know that since our line is perpendicular to the line x = 1 + t , y = 1-t , z = 2 t , it is also perpendicular to the direction vector of that line. To find the direction vector, we simply look at the coefficients in front of the t ’s. Thus, the direction vector for this line is h 1 ,-1 , 2 i . We now have two vectors that our line (or, more specifically, our line’s direction vector) is perpendicular to. Thus, we may simply compute the cross product h 1 , 1 , 1 i × h 1 ,-1 , 2 i to find a suitable direction vector for our line. ± ± ± ± ± ± i j k 1 1 1 1-1 2 ± ± ± ± ± ± = 3 i-1 j-2 k Now let’s plug this into our formula for a parametrized line. When we do, we get x = 3 t, y = 1-t, z = 2-2 t. 1...
View Full Document

This homework help was uploaded on 04/17/2008 for the course MATH 231 taught by Professor Bociu during the Spring '08 term at UVA.

Ask a homework question - tutors are online