MATH 200 Quiz: math200Quiz2bSolutions
MATH 200 104 Quiz 2
First Name: Last Name: Student Number:
1. (2 points) Determine if the following are parallel, perpendicular or neither.
(a) The line 4x=y=−5zand the plane x+y+z= 1.
The direction of the line is ~m =<1/4,1,−1/5>.
The normal vector of the plane is ~n =<1,1,1>.
~m ·~n =<1/4,1,−1/5>·<1,1,1>=1
4+ 1 −1
5=21
20 6= 0.
These two vectors are neither parallel nor perpendicular which implies that the line and
the plane are not parallel and not perpendicular.
Neither
(b) The plane z= 2x−3y+ 1 and the plane z=3−4x+6y
2.
The normal vector of the first plane 0 = 2x−3y−z+ 1 is ~n1=<2,−3,−1>.
The normal vector of the second plane 0 = 3−4x+6y
2−zis ~n2=<−4
2,6
2,−1>.
~n1·~n2=<2,−3,−1>·<−4
2,6
2,−1>=−4−9+16= 0.
These two vectors are neither parallel nor perpendicular which implies that the planes
are not parallel or perpendicular.
Neither
2. (8 points) Consider the line ~r(t) =<2,2,0>+t < −1,−1,3>.
(a) At what point does the plane x+y+z= 3 intersect the line ~r(t)?
We will plug x(t) = −t+ 2, y(t) = −t+ 2 and z(t)=3tinto the plane x+y+z= 3.
(−t+ 2) + (−t+ 2) + (3t)=3
t=−1
x(−1) = −(−1) + 2 = 3
y(−1) = −(−1) + 2 = 3
z(−1) = 3(−1) = −3
(3,3,−3)
(b) What is the equation of the plane which contains the line ~r(t) and the origin?
The direction of the line ~m =<−1,−1,3>is one the plane so is perpendicular to the
normal vector ~n
The point P= (2,2,0) is on the line so is in the plane. The point O= (0,0,0) is also
on the plane. So the vector ~
OP =<2,2,0>is on the plane so is perpendicular to the
normal vector ~n.
~n =~m ×~
OP =<−6,6,0>
Use the normal vector ~n and the point (0,0,0) to define the plane.
−6x+ 6y= 0
(c) What angle does the line ~r(t) make with the x-yplane?
The angle ~r(t) makes with the x-yplane is the angle that the direction ~m =<−1,−1,3>
makes with the x-yplane.
1
51
MATH 200 Full Course Notes
Verified Note
51 documents
Document Summary
Student number: (2 points) determine if the following are parallel, perpendicular or neither. (a) the line 4x = y = 5z and the plane x + y + z = 1. The direction of the line is ~m =< 1/4, 1, 1/5 >. The normal vector of the plane is ~n =< 1, 1, 1 >. ~m ~n =< 1/4, 1, 1/5 > < 1, 1, 1 >= 1. These two vectors are neither parallel nor perpendicular which implies that the line and the plane are not parallel and not perpendicular. 5 = 21 (b) the plane z = 2x 3y + 1 and the plane z = 3. The normal vector of the rst plane 0 = 2x 3y z + 1 is ~n1 =< 2, 3, 1 >. The normal vector of the second plane 0 = 3. These two vectors are neither parallel nor perpendicular which implies that the planes are not parallel or perpendicular.
