Search & Filtering
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.
(3.1) Find the point of intersection between the lines: < 3, −1, 2 > + t < 1, 1, −1 > and
< −8, 2, 0 > + t < −3, 2, −7 >.
(3.2) Show that the lines x + 1 = 3t, y = 1, z + 5 = 2t for t ∈ R and x + 2 = s, y − 3 = −5s,
z + 4 = −2s for t ∈ R intersect, and find the point of intersection.
(3.3) Find the point of intersection between the planes: −5x + y −2z = 3 and 2x −3y + 5z = −7.
(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 of intersection of the planes −5x + y − 2z = 3 and
2x − 3y + 5z = −7.
(1.1) Determine whether the given line and the given plane are parallel:
(a) x = 1 + t, y = −1 − t, z = −2t and x + 2y + 3z − 9 = 0,
(b) < 0, 1, 2 > +t < 3, 2, −1 > and 4x − y + 2z + 1 = 0.
Which one of the following describes the points that satisfy the equation
"(x;y;z) = (-1;2;0) + \\lambda" "(1,3,-2)"
- A plane through the point "(-1,2,0)" perpendicular to the line "r = \\lambda(1,3,-2)"
- A plane through the point "(-1,2,0)" parallel to the one "r = \\lambda(1,3,-2)"
- A sphere centered at "(-1,-3,2)" with a radius 5
- A line through the point "(-1,2,0)" with direction vector "d = \\lambda(1,3,-2)"
1. Find a point-normal form of the equation of the plane passing through P = (1, 2,-3) and having n =< 2,-1, 2 > as a normal.
2. Determine in each case whether the given planes are parallel or perpendicular:
a) x + y + 3z + 10 = 0 and x + 2y - z = 1
b) 3x - 2y + z - 6 = 0 and 4x + 2y - 4z = 0
a) 3x + y + z - 1 = 0 and - x + 2y + z + 3 = 0
b) x - 3y + z + 1 = 0 and 3x - 4y + z - 1 = 0
Question 6
Let u =< −2,1,−1,v =< −3,2,−1 >and w =< 1,3,5 >. Compute:
(6.1) u ×w,
(6.2)u ×(w ×v)and(u × w)×v.
