quiz2soln

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

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

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
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