X x y y z z d1 d2 d3 - UH
Math 2433 Notes ? Week 1 Session 2
12.6 Lines
Vector form: r(t) = (x0i + y0 j + z0k) + t(d1i + d2 j + d3k)
x(t) = x0 + td1 Scalar form: y(t) = y0 + td2
z(t) = z0 + td3
Symmetric form: x - x0 = y - y0 = z - z0
d1
d2
d3
Examples: 1) Find a vector parameterization for the line that passes through P(3, 2, 3) and is parallel to the line r(t) = (i + j - k) + t(2i + 3j - 3k).
2) Give the answer to #1 in scalar form 3) Give the answer to #1 in symmetric form 4) Find a set of scalar parametric equations for the line that passes through P(2, 2, 1) and Q(3, -2, -2).
5) Find a set of scalar parametric equations for the line that passes through P(1, 2, -3) and is perpendicular to the xz-plane.
Popper 1
6.
A direction vector for the line
x-4 -2
=
y+ 3 = z-1 5
is
a. d = (4, -3, 1)
b. d = (-2, 5, 0)
c. d = (2, -5, -1)
d. d = (-2, 5, 1)
e. None of the above
Parallel lines have direction vectors that are scalar multiples of each other. 6) Give a vector parameterization for the line that passes through P(1, 2, -2) and is parallel to the line: 3(x - 1) = 2(y - 2) = 6(z + 2)
7) Determine whether the lines l1 and l2 are parallel, coincident, skew, or intersecting. If they intersect, find the point of intersection:
Two lines (l1: r(t) = r0 + t d and l2: R(u) = R0 + u D) intersect if there exists numbers t and u such that r(t) = R(u) 8) Determine whether the lines l1 and l2 are parallel, coincident, skew, or intersecting. If they intersect, find the point of intersection:
If two lines are not parallel and do not intersect, then they are skew. 9) Determine whether the lines l1 and l2 are parallel, coincident, skew, or intersecting. If they intersect, find the point of intersection:
Popper 1 7. The lines 1 : x = -2 + 4t, y = 3 - 2t, z = 3 + 6t and 2 : x = 6 - 2s, y = -1+ s, z = 3 - 3s are a. Coincident b. Parallel but not coincident c. Have a unique point of intersection d. Skew e. None of the above
To find the angle between two lines:
cos = ud ? uD
Distance from a point to a line: d(P, l) = P0 P1 ? d d
10) Find the distance from P(2, 0, 2) to the line through P0(3, -1, 1) parallel to i - 2j - 2k.
Popper 1
8.
The
cosine of
the
angle between the lines
1
:
x
+ 3
2
=
y-2 -2
=
z +1 -2
and
2 : x = 2 - 3s, y = -4 + s, z = 1+ 4s is
a. -19 26 17
b.
1
26 17
c. 19 26 17
d. 15 26 17
e. None of these
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