1. Find a vector parameterization for the line that passes through P(3, 3, 0) and is parallel to the line r(t) = (i - j) + tk.
a) r2(t)=(i-j)+tk
b) r2(t)=(i-j)+t(3i+3j)
c) r2(t)=(k)+t(i-j)
d) r2(t)=(3i+3j)+tk
e) r2(t)=(k)+t(3i+3j)
2. Find a set of scalar parametric equations for the line that passes through P(3, 4, 1) and Q(4, 0, -1).
a) x(t)=3-2t,y(t)=4-2t,z(t)=1-5t
b) x(t)=1+2t,y(t)=-4-6t,z(t)=1
c) x(t)=3,y(t)=4-2t,z(t)=1-3t
d) x(t)=1-2t,y(t)=-7t,z(t)=-4-3t
e) x(t)=3+t,y(t)=4-4t,z(t)=1-2t
3.Find a set of scalar parametric equations for the line that passes through P(1, 3, -4) and is perpendicular to the xy-plane.
a) x(t)=t,y(t)=3t,z(t)=1-4t
b) x(t)=1-2t,y(t)=3+2t,z(t)=-4
c) x(t)=-3t,y(t)=-3t,z(t)=-4-t
d) x(t)=1,y(t)=3,z(t)=-4+t
e) x(t)=1+t,y(t)=3-t,z(t)=-4+4t
4.Give a vector parametrization for the line that passes through P(2, 2, -3) and is parallel to the line:2(x-2)=(y-2)=4(z+3)
a) (2i + 2j - 3k) + t (2i + 12j - k)
b) (2i + 2j - 3k) + t (2i + 4j + k)
c) (2i + 2j - 3k) + t (2i + 4j - 2k)
d) (2i - 2j + 3k) + t (2i - 4j + k)
e) (2i + 2j - 3k) + t (2i - 4j + 2k)
5. Determine whether the lines l1 and l2 are parallel, coincident, skew, or intersecting. If they intersect, find the point of intersection:
a) Intersect at (-1,3,2)
b) Skew
c) Intersect at (1,3,1)
d) Coincident
e) Parallel but not coincident
6. Determine whether the lines l1 and l2 are parallel, coincident, skew, or intersecting. If they intersect, find the point of intersection:
a) Intersect at (1,0,1)
b) Intersect at (-1,3,2)
c) Parallel but not coincident
d) Coincident
e) Skew
7. Find the distance from P(2, 0, 2) to the line through P0(2, -1, 1) parallel to i - 3j - 3k.
a) 3√3
b) 38--√2
c) 38--√19
d) 114---√6
e) 257--√19
8. A direction vector for the line x-23=y-24=z+33is:
a) d=(2,2,3)
b) d=(-23,12,-1)
c) d=(4,-3,4)
d) d=(24,-24,36)
e) d=(3,4,3)
9. Given l1:r1(t)=(i-6j+2k)+t(i+2j+k),l2:r2(u)=(4j+k)+u(2i+j+2k), find the point where l1 and l2intersect and find the angle between l1 and l2.
a) Point: (4,4,3),θ=π6
b) Point: (8,8,9),θ=π3
c) Point: (4,4,3),θ=π4
d) Point: (8,8,9),θ=cos-1(6√3)
e) Point: (8,8,9),θ=cos-1(6√2)
10. Given l1:r1(t)=(i-6j+2k)+t(i+2j+k),l2:r2(u)=(4j+k)+u(2i+j+2k), find the point where l1 and l2intersect and give the scalar parametric form for the line l3which is perpendicular to both l1 and l2through that point.
a) Point: (4,4,9),l3:x(t)=4-3t,y(t)=4+t,z(t)=9-t
b) Point: (8,8,9),l3:x(t)=8-3t,y(t)=8,z(t)=9-t
c) Point: (4,4,9),l3:x(t)=4-t,y(t)=4+2t,z(t)=9+t
d) Point: (8,8,9),l3:x(t)=8+3t,y(t)=8,z(t)=9-3t
e) Point: (8,8,9),l3:x(t)=8+3t,y(t)=8+t,z(t)=9-3t