Analytic Geometry Answers

Let the plane P pass through the 3 points R = (1, 1, 6), S= (2,5,4) and T = (1,2,3).

Fill in the following:

1. RS = (__,__,__)
2. RT = (__,__,__)
3. RS x RT = (__,__,__)

Given A= i+2j+3k and B= 2i+j-k, find AxB

1.find an equation for the plane that passes through the origin (0,0,0) and is parall to the plane -x+3y-2z=6.

2.find the distance between the point (-1,-2,0) and the plane 3x-y+4z=-2.

3.find the components of a unit vector satisfying V.<3,-1>=0.

1.find the vector form of the equation of the plane that passes through the point po=(1,-2,3) and has normal vector n=<3,1,-1>.

2.find an equation for the plane that contains the line x=-1+3t,y=5+3t,z=2+t and is parallel to the line of intersection of the planes x-2(y-1)+3z=-1 and y=-2x-1=0.

Let L be the line given by <3,-1,2>+t<1,1,-1>,for tER.

1.show that the above line L lies on the plane -2x+3y-4z+1=0.

2.find an equation for the plane through the point p=(3,-2,4) that is perpendicular to the line <-8,2,0>+t<-3,2,-7>.

1.find the point of intersection between the lines:<3,-1,2>+t<1,1,-1> and <-8,2,0>+t<-3,2,-7>.

2.show that the lines x+1=3t,y=1,z+5=2t for tER and z+2=s,y-3=-5s,z+4=-2s for tER intersect, and find the point of intersection.

3.find the point of intersection between the planes: -5x+y-2z=3 and 2x-3y+5z=-7.

1.determine whether the given line and the given planes are parallel:

1.1.x=1+t,y=-1-t,z=-2t and x+2y+3z-9=0.

1.2.<0,1,2>+t<3,2,-1> and 4x-y+2z+1=0.

2.

2.1.find parametric equations of the line that passes through the point p=(2,0,-1) and is parallel to the vector n=<2,1,3>.

2.2.find parametric equations of the line that passes through the points a=(1,2,-3) and b=(7,2,-4).

2.3.find parametric equations for the line intersection of the planes -5x+y-2z=3 and 2x-3y+5z=-7.

Let u =< 3, −1, −2 >, v =< −1, 0, 2 > and w =< −6, 1, 4 >.

Compute the expressions below.

a) >v + 3w

b) 3>u − 2>v (1)

c) −(>v + 3>u). 

Question 5:

Assume that a vector a of length ||a|| = 3 units. In addition, a points in a direction that is 135◦ counterclockwise from the positive x-axis, and a vector b in th xy-plane has a length ||b|| = 1 3 and points in the positive y-direction.

(5.1) Find a ·b.

(5.2) Calculate the distance between the point (−1,√3) and the line 2x-2y-5=0

(5.1) Find the vector form of the equation of the plane that passes through the point P0 = (1, −2, 3)

and has normal vector ~n =< 3, 1, −1 >.

(5.2) Find an equation for the plane that contains the line x = −1 + 3t, y = 5 + 3t, z = 2 + t and is

parallel to the line of intersection of the planes x −2(y −1) + 3z = −1 and y = −2x −1 = 0.