Introduction
Parallel lines
Intersecting lines
Skew lines
Coincident lines
Test
PAIRS OF LINES
VECTORS 2
INU0114/514 (MATHS 1)
Dr Adrian Jannetta MIMA CMath FRAS
Pairs of Lines
1 / 16
Adrian Jannetta
Introduction
Parallel lines
Intersecting lines
Skew lines
Coincident lines
Test
Objectives
The purpose of this presentation is to cover the following topics:
• Be able to show whether two vector lines are parallel or not.
• If two vector lines are not parallel then be able to show
whether the lines intersect or not.
Pairs of Lines
2 / 16
Adrian Jannetta
Introduction
Parallel lines
Intersecting lines
Skew lines
Coincident lines
Test
Pairs of lines
In 2D (Euclidean) geometry a pair of straight lines are either
parallel or, if not, they will intersect somewhere.
Pairs of lines in 3D space do not necessarily intersect when they
aren’t parallel. The location of two lines in 3D space may such that
1
They are parallel and distinct (no intersection)
2
They are parallel and coincident (infinite intersections)
3
They are not parallel and intersect at one point.
4
They are not parallel and don’t intersect.
We call them skew lines in this case.
These possibilities are shown on the following slide.
Pairs of Lines
3 / 16
Adrian Jannetta
Introduction
Parallel lines
Intersecting lines
Skew lines
Test
z
z
L2
L1
L2
L1
y
x
y
x
Coincident lines
Not parallel; one intersection.
Parallel and distinct; no intersections.
z
z
L1
L2
L1
x
y
y
x
L2
Parallel and coincident; infinite intersections.
Pairs of Lines
Not parallel; no intersection (skew).
4 / 16
Adrian Jannetta
Introduction
Parallel lines
Intersecting lines
Skew lines
Coincident lines
Test
Parallel lines
Recall that two vectors are parallel if they satisfy b = ka where k is
a constant.
Consider the two lines
r1
= a1 + λd1
r2
= a2 + µd2
To see if the lines are parallel we just need to check the direction
vectors d1 and d2 .
If the lines are parallel then we would expect to find
d2 = kd1
If lines are parallel they could be distinct (no intersection) or
coincident (infinite intersections).
Pairs of Lines
5 / 16
Adrian Jannetta
Introduction
Parallel lines
Intersecting lines
Skew lines
Coincident lines
Test
Pairs of lines
Show that the lines given by
r1
= 3i + 4j − 2k + λ(−i + 2j − 9k)
r2
= 10i − 2j − 2k + µ(−2i + 4j − 18k)
are parallel.
The direction vectors are
d1
= −i + 2j − 9k
d2
= −2i + 4j − 18k
Comparing the components: d2 = 2d1 .
The lines are parallel.
Pairs of Lines
6 / 16
Adrian Jannetta
Introduction
Parallel lines
Intersecting lines
Skew lines
Coincident lines
Test
Non-parallel lines
Consider the two lines
r1
= a1 + λd1
r2
= a2 + µd2
If the lines are not parallel then can try to find an intersection.
If the lines intersect then there must be unique values of λ and µ
such that
a1 + λd1 = a2 + µd2
i.e. we should be able to solve the equations to find values of λ
and µ at the intersection.
If no values of values of λ and µ can be found then the lines do not
intersect and are said to be skew.
Pairs of Lines
7 / 16
Adrian Jannetta
Introduction
Parallel lines
Intersecting lines
Skew lines
Coincident lines
Test
Pairs of lines
Consider the lines whose equations are
r1
= i − j + 3k + λ(i − j + k)
r2
= 2i + 4j + 6k + µ(2i + j + 3k)
Show that the lines are non-parallel and intersect.
The direction vectors are i − j + k and 2i + j + 3k.
The lines are not parallel because 2i + j + 3k 6= k (i − j + k) .
If the lines intersect then r1 = r2 . Therefore
i − j + 3k + λ(i − j + k) = 2i + 4j + 6k + µ(2i + j + 3k)
In component form:
(1 + λ)i + (−1 − λ)j + (3 + λ)k = (2 + 2µ)i + (4 + µ)j + (6 + 3µ)k
Pairs of Lines
8 / 16
Adrian Jannetta
Introduction
Parallel lines
Intersecting lines
Skew lines
Coincident lines
Test
(1 + λ)i + (−1 − λ)j + (3 + λ)k = (2 + 2µ)i + (4 + µ)j + (6 + 3µ)k
Equating the components on each side gives three equations:
1+λ
=
2 + 2µ
(1)
−1 − λ
=
4+µ
(2)
3+λ
=
6 + 3µ
(3)
We have three equations and only two unknowns.
We can choose any pair of equations to solve — we’ll always be able to
find values of λ and µ.
Using equations (1) and (2)
1+λ
=
2 + 2µ
−1 − λ
=
4+µ
Solve these simultaneously. In this case we find
λ = −3 and µ = −2
Pairs of Lines
9 / 16
Adrian Jannetta
Introduction
Parallel lines
Intersecting lines
Skew lines
Coincident lines
Test
We solved (1) and (2) to get λ = −3 and µ = −2.
Now we need to show these values are consistent with equation
(3). If they are consistent then the lines intersect. If not then the
lines are skew.
Substitute into (3) and show LHS = RHS:
LHS = 3 + (−3) = 0
RHS = 6 + 3(−2) = 0
Since the values are true for (3) then the equations are consistent
and the lines must intersect.
To find the intersection point just substitute λ or µ into one of the
orginal equations. For example:
r1
= i − j + 3k − 3(i − j + k)
= −2i + 2j
The lines intersect at (−2, 2, 0).
Pairs of Lines
10 / 16
Adrian Jannetta
Introduction
Parallel lines
Intersecting lines
Skew lines
Coincident lines
Test
Pairs of lines
Consider the lines whose equations are
r1
= i + k + λ(i + 3j + 4k)
r2
= 2i + 3j + µ(4i − j + k)
Determine whether or not the lines intersect.
Check the direction vectors.
The direction vectors are i + 3j + 4k and 4i − j + k.
The lines are not parallel since 4i − j + k 6= k (i + 3j + 4k).
Now we look for an intersection.
Pairs of Lines
11 / 16
Adrian Jannetta
Introduction
Parallel lines
Intersecting lines
Skew lines
Coincident lines
Test
If the lines intersect then
i + k + λ(i + 3j + 4k) = 2i + 3j + µ(4i − j + k)
In component form:
(1 + λ)i + 3λj + (1 + 4λ)k
=
(2 + 4µ)i + (3 − µ)j + µk
Equating the components on each side leads to the equations
1+λ
=
2 + 4µ
(4)
3λ
=
3−µ
(5)
1 + 4λ
=
µ
(6)
As in the previous example, we choose two equations to solve and verify
with the remaining equation.
Using (4) and (5):
1+λ
=
2 + 4µ
3λ
=
3−µ
Solving these gives µ = 0 and λ = 1.
Pairs of Lines
12 / 16
Adrian Jannetta
Introduction
Parallel lines
Intersecting lines
Skew lines
Coincident lines
Test
We solved (4) and (5) to get µ = 0 and λ = 1.
If these are consistent with (6) then the lines intersect.
Substitute λ and µ into (6)
LHS = 1 + 4(1) = 5
RHS = 0
Since LHS 6= RHS then our values of µ = 0 and λ = 1 are not
consistent.
The lines do not intersect; they are skew lines.
Pairs of Lines
13 / 16
Adrian Jannetta
Introduction
Parallel lines
Intersecting lines
Skew lines
Coincident lines
Test
Coincident lines
Consider the parallel lines








2
−1
0
−2
r1 =  0  + λ  3  and r2 =  6  + µ  6 
1
1
3
2
The lines are parallel because d2 = 2d1 .
Now we’ll check whether the two lines are distinct or coincident.
If the lines are coincident, then the point (0, 6, 3) will be on the line
r1 . We should be able to find a value for λ that corresponds to this
point.
Pairs of Lines
14 / 16
Adrian Jannetta
Introduction
Parallel lines
Intersecting lines
Skew lines
Coincident lines
Test
The parametric equations for the lines are
2−λ = 0
3λ = 6
1+λ = 3
Solving the first equation gives λ = 2.
By inspection, we see that this value is also consistent with the
other two equations.
Since the lines are parallel and pass through the shared point,
then they are coincident lines.
(Note that we could have proved the same result using µ; by
showing that (2, 0, 1) is on r2 ).
Pairs of Lines
15 / 16
Adrian Jannetta
Introduction
Parallel lines
Intersecting lines
Skew lines
Coincident lines
Test
Test yourself
If you’ve read and understood the examples in these notes, you should
be able to answer the following questions.
Decide if the following pairs of lines are parallel or not. If not, are they
intersecting or skew?
1
r = −2i + j + λ(2i − j) and r = i + j + µ(8i − 4j)
2
r = 2i − k + λ(−2i + 4j − k) and r = 3i − 5j − 5k + µ(i − 3j − k)
3
r = 5i − 4j − 2k + λ(i + 2j + 3k) and r = 2i + k + µ(2i − j − k)
4
r = 3i + j + 2k + λ(i − j + 2k) and r = 6i − 2j + 8k + µ(−i + j − 2k)
1 Parallel and distinct.
2 Not parallel; intersection at (0, 4, −2).
3 Not parallel; skew.
4 Parallel and coincident.
Pairs of Lines
16 / 16
Adrian Jannetta