1
00:00:06,844 --> 00:00:09,010
DAVID JORDAN: Hello, and
welcome back to recitation.

2
00:00:09,010 --> 00:00:10,801
The problem I'd like
to work with you today

3
00:00:10,801 --> 00:00:14,060
is the intersection of
two parametrized lines.

4
00:00:14,060 --> 00:00:15,910
So we have two lines here.

5
00:00:15,910 --> 00:00:18,670
L_1, given with
the parametrization

6
00:00:18,670 --> 00:00:20,820
in terms of the variable t.

7
00:00:20,820 --> 00:00:23,950
And L_2, also given
with the parametrization

8
00:00:23,950 --> 00:00:25,570
in terms of the variable t.

9
00:00:25,570 --> 00:00:29,390
So the first question
that I want us to answer

10
00:00:29,390 --> 00:00:32,250
is do these lines intersect?

11
00:00:32,250 --> 00:00:36,267
And if so, then we want to find
out where do they intersect.

12
00:00:36,267 --> 00:00:38,350
So why don't you pause the
video and work on this.

13
00:00:38,350 --> 00:00:40,060
And we can check
back in a moment

14
00:00:40,060 --> 00:00:41,310
and we'll see how I solved it.

15
00:00:49,230 --> 00:00:49,980
OK, welcome back.

16
00:00:49,980 --> 00:00:52,930
Let's get started.

17
00:00:52,930 --> 00:00:54,920
So we have these
two lines in space.

18
00:00:54,920 --> 00:00:57,252
Before we start doing
any computations,

19
00:00:57,252 --> 00:00:58,710
I find it useful
to draw a picture.

20
00:00:58,710 --> 00:01:00,190
So let's see what's going on.

21
00:01:06,950 --> 00:01:07,450
OK.

22
00:01:07,450 --> 00:01:09,730
So we have these two lines.

23
00:01:09,730 --> 00:01:15,150
We can just find some
common points on the lines.

24
00:01:15,150 --> 00:01:17,720
So, well, if we put
in t equals 0 here,

25
00:01:17,720 --> 00:01:20,780
then it looks like we
get the point 2 comma 1.

26
00:01:26,820 --> 00:01:27,650
OK.

27
00:01:27,650 --> 00:01:33,980
And now if we plug in, let's
say, t is minus 1 here,

28
00:01:33,980 --> 00:01:41,920
then we get-- if we plug in t
is minus 1 here, we get x is 3

29
00:01:41,920 --> 00:01:42,960
and y is 0.

30
00:01:48,350 --> 00:01:49,420
So there's our line L_1.

31
00:01:52,320 --> 00:01:54,380
And now let's see, L_2.

32
00:01:54,380 --> 00:02:05,260
If we plug in t equals 0,
we get 2-- 1, 2, 3-- 4.

33
00:02:05,260 --> 00:02:10,670
And if we plug in, let's
say, t equals minus 1 again,

34
00:02:10,670 --> 00:02:19,215
then we get 1 and 2.

35
00:02:24,471 --> 00:02:24,970
OK?

36
00:02:24,970 --> 00:02:27,500
So there is L_2.

37
00:02:27,500 --> 00:02:31,927
And indeed, it does look
like they intersect.

38
00:02:31,927 --> 00:02:34,010
We could have probably
guessed that they intersect

39
00:02:34,010 --> 00:02:37,940
by looking back over here at
the formulas for L_1 and L_2,

40
00:02:37,940 --> 00:02:40,850
because we see that the
sort of direction that this

41
00:02:40,850 --> 00:02:44,150
is moving in, we can
take derivatives in t.

42
00:02:44,150 --> 00:02:47,520
And we see that L_2
is, this line is moving

43
00:02:47,520 --> 00:02:50,850
in the direction 1 comma 2.

44
00:02:50,850 --> 00:02:54,490
And L_1 is moving in the
direction minus 1 comma 1.

45
00:02:54,490 --> 00:02:57,484
And so those directions
are not parallel.

46
00:02:57,484 --> 00:02:59,150
And so we know that
the only way the two

47
00:02:59,150 --> 00:03:01,805
lines could fail to intersect
is if they're parallel.

48
00:03:01,805 --> 00:03:03,430
So actually, even
without drawing this,

49
00:03:03,430 --> 00:03:05,596
we could have guessed that
these lines do intersect.

50
00:03:07,780 --> 00:03:09,530
So now we know that
these lines intersect.

51
00:03:09,530 --> 00:03:11,440
And in fact, it even
looks, you know,

52
00:03:11,440 --> 00:03:15,510
it kind of looks like they
intersect-- from our sketch--

53
00:03:15,510 --> 00:03:19,990
at the point-- it
looks like-- 1 comma 2.

54
00:03:19,990 --> 00:03:21,970
It seems to be the
point of intersection.

55
00:03:21,970 --> 00:03:26,899
But, you know, we got a little
bit lucky with our sketch here.

56
00:03:26,899 --> 00:03:28,440
So let's see if
that's actually true.

57
00:03:28,440 --> 00:03:30,542
Let's see if we can verify
this in the general way

58
00:03:30,542 --> 00:03:31,750
that we discussed in lecture.

59
00:03:35,809 --> 00:03:38,100
So, now there is one place
where we have to be careful.

60
00:03:38,100 --> 00:03:43,320
We have two lines here, and
we parametrized both of these

61
00:03:43,320 --> 00:03:44,285
with the variable t.

62
00:03:44,285 --> 00:03:46,850
But we need to keep
in mind that t is

63
00:03:46,850 --> 00:03:49,130
what's called a dummy variable.

64
00:03:49,130 --> 00:03:51,920
It doesn't have any geometric
meaning to the problem.

65
00:03:51,920 --> 00:03:56,200
And in particular, what I
want to caution you about

66
00:03:56,200 --> 00:03:59,460
is if we just start
solving these two

67
00:03:59,460 --> 00:04:01,870
equations algebraically
as they're

68
00:04:01,870 --> 00:04:06,360
given to us with the
variable t, the problem

69
00:04:06,360 --> 00:04:09,460
that we could run into
is that, you know,

70
00:04:09,460 --> 00:04:11,830
we're sort of
moving-- as we vary

71
00:04:11,830 --> 00:04:15,182
t-- we're moving along this
line, and as we vary t again,

72
00:04:15,182 --> 00:04:16,390
we're moving along this line.

73
00:04:16,390 --> 00:04:20,519
And you see we're
moving at the same time.

74
00:04:20,519 --> 00:04:22,560
And so that's really
solving a different problem.

75
00:04:22,560 --> 00:04:24,972
That's not asking about when
did these lines intersect,

76
00:04:24,972 --> 00:04:26,430
but that would be
asking about when

77
00:04:26,430 --> 00:04:29,270
do two particles on these
lines collide, which

78
00:04:29,270 --> 00:04:30,600
is a harder problem.

79
00:04:30,600 --> 00:04:33,200
So instead, what
we need to do is

80
00:04:33,200 --> 00:04:38,000
we need to give a change of
variables for the line L_2.

81
00:04:38,000 --> 00:04:43,440
So what I want to do is I'm
going to write L_2-- I'm just

82
00:04:43,440 --> 00:04:47,360
going to write the
same equations,

83
00:04:47,360 --> 00:04:50,000
but I'm going to introduce
a new variable u.

84
00:04:50,000 --> 00:04:55,790
So x is 2 plus u,
and y is 4 plus 2u.

85
00:04:58,980 --> 00:04:59,480
OK.

86
00:04:59,480 --> 00:05:02,760
So now once we've done that, to
find the point of intersection,

87
00:05:02,760 --> 00:05:05,560
well, the point
of intersection is

88
00:05:05,560 --> 00:05:08,170
going to precisely
be a point on L_2

89
00:05:08,170 --> 00:05:10,590
where the x-coordinate
and the y-coordinate

90
00:05:10,590 --> 00:05:14,677
agree with another point
on L_1 with the same

91
00:05:14,677 --> 00:05:15,510
x- and y-coordinate.

92
00:05:15,510 --> 00:05:19,620
So that is, we
have the-- what we

93
00:05:19,620 --> 00:05:24,450
want to do is we want to set
the x-coordinate for L_2.

94
00:05:24,450 --> 00:05:28,310
We want to set this equal to
the x-coordinate for L_1, which

95
00:05:28,310 --> 00:05:30,895
is 2 minus t.

96
00:05:30,895 --> 00:05:33,540
So this was for L_1.

97
00:05:33,540 --> 00:05:38,410
And similarly here, we want to
set the y-coordinate for L_2

98
00:05:38,410 --> 00:05:42,842
equal to the
y-coordinate for L_1.

99
00:05:42,842 --> 00:05:44,300
So now if you think
about it, if we

100
00:05:44,300 --> 00:05:46,850
can solve this system of
equations, then what we've done

101
00:05:46,850 --> 00:05:48,850
is we've simultaneously
found a point which

102
00:05:48,850 --> 00:05:52,747
is on L_1 and on L_2.

103
00:05:52,747 --> 00:05:53,580
And that's our goal.

104
00:05:53,580 --> 00:05:56,000
So that will be a
point of intersection.

105
00:05:56,000 --> 00:05:58,520
OK, so now we just
have this system

106
00:05:58,520 --> 00:06:00,270
of two linear equations
and two variables,

107
00:06:00,270 --> 00:06:01,900
and we just need to solve it.

108
00:06:01,900 --> 00:06:07,232
Now, we could do, you
know, the-- in general,

109
00:06:07,232 --> 00:06:09,690
with an equation like this, we
might try to add or subtract

110
00:06:09,690 --> 00:06:10,530
the equations.

111
00:06:10,530 --> 00:06:12,620
But this one is
so simple, that I

112
00:06:12,620 --> 00:06:22,560
see that the top equation
is just the same thing as t

113
00:06:22,560 --> 00:06:24,270
equaling to minus u.

114
00:06:24,270 --> 00:06:27,270
That's what the top equation
says if we cancel the 2's.

115
00:06:27,270 --> 00:06:30,800
And so if we plug that
into the next equation,

116
00:06:30,800 --> 00:06:37,310
then we get 4 plus
2u equals 1 minus u.

117
00:06:37,310 --> 00:06:39,887
And so then we can
solve this, and we get,

118
00:06:39,887 --> 00:06:48,000
so it looks like 3 equals
minus 3u, which tells us

119
00:06:48,000 --> 00:06:52,650
that u equals minus
1, and then that

120
00:06:52,650 --> 00:06:55,700
tells us that t equals plus 1.

121
00:06:59,260 --> 00:07:00,850
OK?

122
00:07:00,850 --> 00:07:02,840
So we found our
parameters t and u.

123
00:07:02,840 --> 00:07:04,290
And we're not quite done yet.

124
00:07:04,290 --> 00:07:10,450
What we need to do is we need to
go back to our parametrization.

125
00:07:10,450 --> 00:07:15,000
So let me go back over to our
original parametrization here,

126
00:07:15,000 --> 00:07:20,590
and we have L_1 was 2
minus t and 1 plus t.

127
00:07:20,590 --> 00:07:22,560
And over here, we
found that t equals 1

128
00:07:22,560 --> 00:07:24,160
was the value that we're after.

129
00:07:24,160 --> 00:07:37,420
So that tells us that x is
1-- 2 minus 1-- and y is 2.

130
00:07:37,420 --> 00:07:38,594
Excuse me.

131
00:07:38,594 --> 00:07:39,760
I wrote that in a funny way.

132
00:07:42,889 --> 00:07:47,850
x is 1 and y is 2.

133
00:07:47,850 --> 00:07:49,135
Now, just as a reality check.

134
00:07:52,200 --> 00:07:56,040
We also found that
if we solved for L_2,

135
00:07:56,040 --> 00:07:59,170
we wanted the variable u
to be equal to minus 1.

136
00:08:02,090 --> 00:08:08,699
So we had 2 plus
u, and 4 plus 2u.

137
00:08:08,699 --> 00:08:10,240
And so let's see
what happens when we

138
00:08:10,240 --> 00:08:13,140
plug in u equals minus 1 here.

139
00:08:13,140 --> 00:08:19,180
We again get x equals
2 plus minus 1 is 1.

140
00:08:19,180 --> 00:08:24,370
And y equals 4
plus minus 2 is 2.

141
00:08:24,370 --> 00:08:27,790
So we just double check that
this is a point of intersection

142
00:08:27,790 --> 00:08:29,660
of both lines.

143
00:08:29,660 --> 00:08:31,545
And I'll leave it at that.