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.