Post on 01-Jan-2016
Specialist Maths
Vectors and Geometry
Week 4
Lines in Space
111 ,,A zyx
cba ,,
cbatzyxzyx ,,,,,, 111
zyx ,,P
Vector Equation
Parametric Equations
c
zz
b
yy
a
xx 111
Cartesian Equation
ctzz
btyy
atxx
1
1
1
Example 21 (Ex 4I)
forms.different 3 in the
equation the Write.654 ofdirection the
in 321 through line theofequation theFind
kji
,,-
Solution 21
ctzz
btyy
atxx
1
1
1
c
zz
b
yy
a
xx 111
cbatzyxzyx ,,,,,, 111
forms.different 3 in the
equation the Write.654 ofdirection the
in 321 through line theofequation theFind
kji
,,-
Example 22 (Ex 4I)
.5 coordinate- a with line on thepoint theFind (c)
line. theof direction in the vector a Find (b)
form. parametricin 1
3
2
12 Write(a)
z
zyx
Solution 22
.5 coordinate- a with line on thepoint theFind (c)
line. theof direction in the vector a Find (b)
form. parametricin 1
3
2
12 Write(a)
z
zyx
Example 23 (Ex 4I)
form. parametricin 201B and
412A through line theofequation theFind
,,-
,,-
Solution 23
form. parametricin 201B and
412A through line theofequation theFind
,,-
,,-
Example 24 (Ex 4I)
plane. YOZ thecuts 23
and 5 ,24 line thehere Find
tz
tyt-xw
Solution 24
plane. YOZ thecuts 23
and 5 ,24 line thehere Find
tz
tyt-xw
Example 25 (Ex 4I)
.3 and
34 ,21 line the to1,2,3-P from
normal theoffoot theof scoordinate theFind
tz
tytx
Solution 25
.3 and
34 ,21 line the to1,2,3-P from
normal theoffoot theof scoordinate theFind
tz
tytx
Shortest distance from a point to a line
a
N
P
A
d
a
aAPd
Example 26 (Ex 4I)
.3 and 34 ,21 line the to
1,2,3-P from distance normal normal theFind
tztytx
a
aAPd
Solution 26
.3 and 34 ,21 line the to
1,2,3-P from distance normal normal theFind
tztytx
a
aAPd
Parallel Lines are always in the same plane and don’t intersect
l1
l2
Angle between parallel lines is zero
Intersecting lines will always be on the same plane
θ
l1
l2
To prove lines are in the same plane you show either that theyIntersect or they are parallel.
Angle between l1 and l2 is θ
Skew Lines are lines that are not parallel but do not meet. They lie in non parallel planes.
Line 1
Line 2
Non parallel planes
Skew Lines
Example 27 (Ex 4I)
2
31 :2 line
41 and 21,21 :1 line
parallel. are lines following that theShow
-zy-x
tztytx
Solution 27
2
31 :2 line
41 and 21,21 :1 line
parallel. are lines following that theShow
-zy-x
tztytx
Example 28 (Ex 4I)
3
41
2
1-x :2 line
23 and ,1 :1 line
intersect. lines following that theShow
z
y
tztytx
Solution 28
3
41
2
1-x :2 line
23 and ,1 :1 line
intersect. lines following that theShow
z
y
tztytx
Example 29 (Ex 4I)
3
41
2
1-x :2 line
41 and 21,21 :1 line
skew. are lines following that theShow
z
y
tztytx
Solution 29
3
41
2
1-x :2 line
41 and 21,21 :1 line
skew. are lines following that theShow
z
y
tztytx
Acute Angle between Lines
A
B
u
v
Line 1
Line 2
Line 1´
Pθ
a
b
ba
ba cos
Example 30 (Ex 4I)
szsysx
tztytx
4 and 21,3 :2 line
2 and 1, :1 line
lines. skew thebetween angle acute theFind
Solution 30
szsysx
tztytx
4 and 21,3 :2 line
2 and 1, :1 line
lines. skew thebetween angle acute theFind
Shortest Distance between Skew Lines
A
B
u
v
Line 1
Line 2P
θ
a
b
ba
bad
AB
Example 31 (Ex 4I)
szsysx
tztytx
4 and 21,3 :2 line
2 and 1, :1 line
lines. skew thebetween distance theFind
ba
bad
AB
Solution 31
szsysx
tztytx
4 and 21,3 :2 line
2 and 1, :1 line
lines. skew thebetween distance theFind ba
bad
AB
This Week
• Text book pages 156 to 154.
• Exercise 4I Questions: 1 – 15.