a
Middle
School
Grade 6
Grade 7
Grade 8
Grade 9
Ressource
Out Middle
High
School
Grade 10
Grade 11
Grade 11 - STMG
Grade 12 - STMG
Grade 12
Grade 12 - Comp.
Grade 12 - Exp.
Ressource
Out High
4
3
2
1
Grade 12
/ Orthogonality, distance in space
60 exercises (including 47 corrected)
a
Lines and orthogonality
(2 exercices)
Lines, planes and orthogonality
(3 exercices)
Mid-plane
(2 exercices)
Distance in space
(4 exercices)
Scalar product definition
(2 exercices)
Bilinearity formula
(2 exercices)
Scalar product calculation
(2 exercices)
Scalar product and orthogonality
(3 exercices)
Determine the measure of an angle
(6 exercices)
Vectors normal to a plane
(3 exercices)
Determine a normal vector to a plane
(4 exercices)
Orthogonal projection on a plane
(10 exercices)
Distance to a plane
(3 exercices)
Orthogonal projection on a straight line
(3 exercices)
Distance from a straight line
(5 exercices)
Scalar product and vector relation
(1 exercice)
image PNG : /home/_math/_exercice/d10/10386/image/Untitled.png
A
B
C
D
E
F
G
H
A
B
C
D
E
F
G
H
I
J
M
N
A
B
C
D
E
F
G
H
A
B
C
D
E
F
G
H
A
B
C
D
I
1.
Lines
and
ortho
gonality
E.5398
In
space,
consider
the
cub
e
AB
C
D
E
F
GH
sho
wn
b
elo
w
:
1
Justify
that
the
straigh
t
lines
(
E
F
)
and
(
GC
)
are
orthog-
onal.
2
Justify
that
the
straigh
t
lines
(
AD
)
and
(
H
F
)
are
not
orthogonal.
3
Are
the
straigh
t
lines
(
AG
)
and
(
B
G
)
orthogonal?
E.6815
The
figure
b
elo
w
sho
ws
a
AB
C
D
E
F
GH
cub
e.
The
p
oin
ts
I
and
J
are
the
re-
sp
ectiv
e
middles
of
the
edges
[
GH
]
and
[
F
G
]
.
P
oin
ts
M
and
N
are
the
resp
ectiv
e
cen
ters
of
faces
AB
F
E
and
B
C
GF
.
Of
the
follo
wing
prop
ositions,
only
one
is
correct.
Whic
h
one?
a
The
straigh
t
lines
(
I
J
)
and
(
M
N
)
are
p
erp
endicular.
b
The
straigh
t
lines
(
I
J
)
and
(
M
N
)
are
secan
t,
not
p
er-
p
endicular.
c
The
straigh
t
lines
(
I
J
)
and
(
M
N
)
are
orthogonal.
d
The
straigh
t
lines
(
I
J
)
and
(
M
N
)
are
parallel.
Justify
y
our
answ
er.
2.
Lines,
planes
and
ortho
gonality
E.576
Consider
the
cub
e
AB
C
D
E
F
GH
with
side
5
cm
sho
wn
b
elo
w
:
1
Sho
w
that
the
plane
(
AB
C
)
is
orthogonal
to
the
line
(
AE
)
2
Calculate
the
length
of
the
diagonal
[
E
C
]
.
E.5399
In
space,
consider
the
cub
e
AB
C
D
E
F
GH
sho
wn
b
elo
w
:
1
a
Justify
that
(
AF
)
is
orthogonal
to
the
plane
(
E
B
C
)
.
b
Deduce
that
(
AF
)
and
(
E
C
)
are
orthogonal.
2
a
Justify
that
(
H
F
)
is
orthogonal
to
the
plane
(
E
C
G
)
.
b
Deduce
that
(
H
F
)
and
(
E
C
)
are
orthogonal.
3
Deduce
that
(
E
C
)
is
orthogonal
to
the
plane
(
AF
H
)
.
E.5397
The
figure
opp
osite
repre-
sen
ts
a
regular
tetrahedron
AB
C
D
and
I
the
middle
of
segmen
t
[
B
D
]
.
Justify
that
the
line
(
B
D
)
is
orthogonal
to
the
plane
(
AI
C
)
.
3.
Mid-plane
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5398
sacados/5398
A
B
C
D
E
F
G
H
chapExoCorrec/6815
sacados/6815
A
B
C
D
E
F
G
H
I
J
M
N
chapExoCorrec/576
sacados/576
A
B
C
D
E
F
G
H
chapExoCorrec/5399
sacados/5399
A
B
C
D
E
F
G
H
chapExoCorrec/5397
sacados/5397
A
B
C
D
I
A
B
C
D
I
J
A
B
C
D
E
F
I
j
k
i
O
S
F
G
E
E.6966
In
space,
consider
the
regular
tetra-
hedron
AB
C
D
sho
wn
b
elo
w
:
P
oin
ts
I
and
J
are
the
middles
of
segmen
ts
[
AD
]
and
[
B
C
]
,
resp
ectiv
ely
.
1
a
Justify
that
the
plane
(
J
AD
)
is
the
median
plane
of
the
segmen
t
[
B
C
]
.
b
What
is
the
relativ
e
p
osition
of
the
straigh
t
lines
(
B
C
)
and
(
I
J
)
?
2
Justify
that
the
straigh
t
lines
(
AD
)
and
(
I
J
)
are
p
erp
en-
dicular.
3
Are
the
straigh
t
lines
(
AD
)
and
(
B
C
)
parallel?
Justify
y
our
answ
er.
E.6868
Consider
a
solid
AD
E
C
B
F
made
up
of
t
w
o
iden
tical
p
yramids
whose
common
base
is
the
square
AB
C
D
of
cen
ter
I
.
A
p
ersp
ectiv
e
represen
tation
of
this
solid
is
giv
en
b
elo
w.
All
edges
are
of
length
1
.
1
Justify
that
the
straigh
t
lines
(
D
E
)
and
(
F
B
)
are
paral-
lel.
2
Justify
that
the
planes
(
AB
F
)
and
(
C
E
D
)
are
parallel.
4.
Distanc
e
in
sp
ac
e
E.2963
Reminders
:
In
a
space
equipp
ed
with
a
co
ordinate
sys-
tem,
let
A
and
B
b
e
t
w
o
p
oin
ts
and
I
b
e
the
midp
oin
t
of
the
segmen
t
[
AB
]
:
AB
=
x
B
−
x
A
2
+
y
B
−
y
A
2
+
z
B
−
z
A
2
I
x
A
+
x
B
2
;
y
A
+
y
B
2
;
z
A
+
z
B
2
In
a
space
equipp
ed
with
an
orthonormal
co
ordinate
system
O
;
I
;
J
;
K
,
w
e
consider
four
p
oin
ts
iden
tified
b
y
their
co-
ordinates
:
A
3
;
−
1
;
5
;
B
−
2
;
2
;
3
;
C
−
1
;
−
2
;
4
;
D
1
2
;
1
2
;
4
1
Are
the
p
oin
ts
A
,
B
,
and
C
aligned?
2
Sho
w
that
the
triangle
AB
C
is
isosceles
at
C
.
3
Justify
that
p
oin
t
D
is
the
fo
ot
of
the
heigh
t
of
triangle
AB
C
from
v
ertex
C
.
E.914
In
space
pro
vided
with
the
ref-
erence
frame
O
;
I
;
J
;
K
or-
thonormal,
consider
the
sphere
S
of
cen
ter
O
and
whose
radius
has
v
alue
2
:
Consider
the
p
oin
ts
E
and
F
whose
co
ordinates
are
:
E
0.96
;
1.28
;
1.2
F
1.2
;
−
√
15
5
;
1.4
1
Sho
w
that
the
p
oin
ts
E
and
F
are
p
oin
ts
on
the
sphere
S
.
2
Let
G
b
e
the
p
oin
t
diametrically
opp
osite
the
p
oin
t
F
in
the
sphere
S
.
Justify
that
the
triangle
E
F
G
is
righ
t-
angled.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/6966
sacados/6966
A
B
C
D
I
J
chapExoCorrec/6868
sacados/6868
A
B
C
D
E
F
I
chapExoCorrec/2963
sacados/2963
chapExoCorrec/914
sacados/914
j
k
i
O
S
F
G
E
A
B
C
D
E
F
G
H
I
J
K
L
M
N
I
A
B
C
D
E
F
G
H
E.2777
Consider
the
space
pro
vided
with
an
orthonormal
reference
frame
(
O
;
I
;
J
;
K
)
.
Consider
the
three
p
oin
ts
A
,
B
,
C
defined
b
y
their
co
ordinates
:
A
180
;
153
;
96
;
B
180
;
135
;
120
;
C
190
;
133
;
106
1
Sho
w
that
the
p
oin
ts
A
,
B
,
C
b
elong
to
the
same
sphere
S
with
cen
ter
O
.
2
Establish
that
the
triangle
AB
C
is
righ
t-angled
C
.
3
a
Is
one
of
the
sides
of
the
triangle
AB
C
a
diameter
of
the
sphere
S
?
b
Can
y
ou
name
a
prop
ert
y
of
the
plane
that
cannot
b
e
extended
to
space?
E.6752
In
space,
consider
the
cub
e
AB
C
D
E
F
GH
of
edge
1
.
The
space
is
giv
en
the
reference
frame
C
;
−
−
→
C
B
;
−
−
→
C
D
;
−
−
→
C
G
or-
thonormal.
The
p
oin
ts
I
,
J
,
K
,
L
,
M
,
N
are
the
resp
ectiv
e
middles
of
the
segmen
ts
[
AB
]
,
[
B
C
]
,
[
C
G
]
,
[
GH
]
,
[
H
E
]
,
[
E
A
]
.
1
Determine
the
co
ordinates
of
p
oin
ts
I
,
K
and
L
.
2
a
Determine
the
co
ordinates
of
p
oin
t
O
midp
oin
t
of
segmen
t
[
I
L
]
.
b
Determine
lengths
O
K
and
K
L
.
3
W
e
admit
that
the
p
olygon
I
J
K
LM
N
is
a
regular
hexagon.
Th
us,
the
p
oin
t
O
is
the
cen
ter
of
this
p
oly-
gon.
a
Giv
e
the
measure
of
the
angle
∠
K
O
L
.
b
Determine
the
area
of
triangle
K
O
L
.
5.
Sc
alar
pr
o
duct
definition
E.8891
Definition
:
let
−
→
u
and
−
→
v
b
e
t
w
o
space
v
ectors.
The
scalar
pro
duct
of
the
v
ectors
−
→
u
and
−
→
v
,
noted
−
→
u
·
−
→
v
,
is
defined
b
y
the
scalar
pro
duct
−
−
→
AB
·
−
→
AC
where
:
the
v
ector
−
−
→
AB
(r
esp.
−
→
AC
)
is
a
represen
tativ
e
of
the
v
ector
−
→
u
(r
esp.
−
→
v
)
.
the
scalar
pro
duct
−
−
→
AB
·
−
→
AC
is
calculated
in
a
plane
con-
taining
the
p
oin
ts
A
,
B
,
C
.
Prop
osition
:
b
e
A
,
B
,
C
three
p
oin
ts
defining
the
t
w
o
non-zero
v
ectors
−
−
→
AB
and
−
→
AC
.
The
scalar
pro
duct
−
−
→
AB
and
−
→
AC
has
the
v
alue
:
−
−
→
AB
·
−
→
AC
=
AB
×
AC
×
cos
∠
B
AC
Consider
the
AB
C
D
E
F
GH
cub
e
with
side
1
sho
wn
op-
p
osite.
P
oin
t
I
is
the
middle
of
seg-
men
t
[
AB
]
.
1
Determine
the
v
alue
of
the
scalar
pro
duct
−
−
→
F
E
·
−
−
→
F
B
2
Determine
the
scalar
pro
duct
:
−
→
I
A
·
−
→
I
B
3
a
What
is
the
nature
of
the
triangle
AC
H
?
b
Deduce
the
v
alue
of
the
scalar
pro
duct
−
−
→
H
A
·
−
−
→
H
C
?
h
ttps:/
/c
hingmath.fr
chapExoCorrec/2777
sacados/2777
chapExoCorrec/6752
sacados/6752
A
B
C
D
E
F
G
H
I
J
K
L
M
N
chapExoCorrec/8891
sacados/8891
I
A
B
C
D
E
F
G
H
A
B
C
D
E
F
G
H
I
J
A
B
C
D
E
F
G
H
I
J
H
G
F
E
D
C
B
A
I
J
E.8892
Prop
osition
:
Let
−
→
u
and
−
→
v
b
e
t
w
o
non-zero
v
ectors
in
the
space.
The
v
ectors
−
→
u
and
−
→
v
are
orthogonal
to
eac
h
other
if
and
only
if
−
→
u
·
−
→
v
=
0
If
the
v
ectors
−
→
u
and
−
→
v
are
collinear
with
eac
h
other
:
−
→
u
and
−
→
v
ha
v
e
the
same
direction
:
−
→
u
·
−
→
v
=
−
→
u
×
−
→
v
−
→
u
and
−
→
v
ha
v
e
opp
osite
directions
:
−
→
u
·
−
→
v
=
−
−
→
u
×
−
→
v
In
space,
consider
the
cub
e
AB
C
D
E
F
GH
with
side
length
1
sho
wn
opp
osite,
where
p
oin
ts
I
and
J
are
the
midp
oin
ts
of
seg-
men
ts
[
GH
]
and
[
AD
]
,
resp
ec-
tiv
ely
.
Using
the
prop
erties
of
the
cub
e
and
the
square,
determine
the
v
alue
of
the
scalar
pro
ducts
:
a
−
−
→
E
H
·
−
−
→
D
H
b
−
−
→
AB
·
−
−
→
D
C
c
−
−
→
B
C
·
−
−
→
GF
6.
Biline
arity
formula
E.5410
Prop
osition
:
Let
−
→
u
,
−
→
v
,
−
→
w
,
and
k
b
e
real
n
um
b
ers.
W
e
ha
v
e
the
follo
wing
iden
tities
:
−
→
u
·
−
→
v
=
−
→
v
·
−
→
u
−
→
u
·
−
→
v
+
−
→
w
=
−
→
u
·
−
→
v
+
−
→
u
·
−
→
w
−
→
u
·
k
·
−
→
v
=
k
·
−
→
u
·
−
→
v
=
k
−
→
u
·
−
→
v
In
space,
consider
the
cub
e
AB
C
D
E
F
GH
with
side
length
1
sho
wn
opp
osite,
where
p
oin
ts
I
and
J
are
the
midp
oin
ts
of
seg-
men
ts
[
GH
]
and
[
AD
]
,
resp
ec-
tiv
ely
.
Metho
d
:
In
calculating
the
scalar
pro
duct,
w
e
use
Chasles’
relation
to
decomp
ose
the
t
w
o
v
ectors
in
to
v
ectors
that
are
collinear
or
orthogonal
to
eac
h
other.
This
facilitates
the
calculation
of
scalar
pro
ducts.
Here
is
an
example
:
T
o
calculate
the
scalar
pro
duct
−
→
I
E
·
−
−
→
GF
,
w
e
use
the
decom-
p
osition
of
v
ector
−
→
I
E
:
−
→
I
E
=
−
→
I
H
+
−
−
→
H
E
W
e
can
then
write
:
−
→
I
E
·
−
−
→
GF
=
−
→
I
H
+
−
−
→
H
E
·
−
−
→
GF
=
−
→
I
H
·
−
−
→
GF
+
−
−
→
H
E
·
−
−
→
GF
=
0
+
H
E
×
GF
=
0
+
1
×
1
=
1
Using
Chasles’
relation
as
w
ell,
determine
the
v
alue
of
the
follo
wing
scalar
pro
ducts
:
a
−
→
AF
·
−
−
→
H
G
b
−
→
J
F
·
−
−
→
AB
c
−
→
I
J
·
−
−
→
E
F
E.4188
In
this
exercise,
an
answ
er
b
y
ˇ
VRAIı
or
ˇ
F
A
UXı,
without
justification,
is
requested
from
the
candidate
against
a
list
of
assertions.
An
y
mathemati-
cally
correct
answ
er
is
w
orth
0.4
p
oin
t.
An
y
incorrect
answ
er
deducts
0.1
p
oin
t.
No
answ
er
is
not
coun
ted.
The
total
cannot
b
e
negativ
e.
W
e
giv
e
the
cub
e
AB
C
D
E
F
GH
,
of
edge
length
1
,
and
the
middles
I
and
J
of
the
edges
[
AB
]
and
[
C
G
]
.
The
useful
ele-
men
ts
of
the
figure
are
giv
en
opp
osite.
The
candidate
is
ask
ed
to
judge
eac
h
of
the
follo
wing
10
state-
men
ts.
Affirmation
V
rai
F
aux
1
−
→
AC
·
−
→
AI
=
1
2
2
−
→
AC
·
−
→
AI
=
−
→
AI
·
−
−
→
AB
3
−
−
→
AB
·
−
→
I
J
=
−
−
→
AB
·
−
→
I
C
4
−
−
→
AB
·
−
→
I
J
=
AB
×
I
C
×
cos
ı
3
7.
Sc
alar
pr
o
duct
c
alculation
E.8890
In
space
with
an
orthonormal
refer-
ence
frame.
1
Consider
the
t
w
o
v
ectors
−
→
u
1
;
2
;
3
and
−
→
v
2
;
−
1
;
1
,
determine
the
scalar
pro
duct
−
→
u
·
−
→
v
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8892
sacados/8892
A
B
C
D
E
F
G
H
I
J
chapExoCorrec/5410
sacados/5410
A
B
C
D
E
F
G
H
I
J
chapExoCorrec/4188
sacados/4188
H
G
F
E
D
C
B
A
I
J
chapExoCorrec/8890
sacados/8890
A
B
C
D
E
F
G
H
−
i
−
k
−
j
I
J
K
A
B
C
D
E
F
G
H
A
B
C
D
E
F
G
H
2
Consider
the
t
w
o
v
ectors
−
→
w
−
2
;
0
;
1
and
−
→
t
−
1
;
1
;
−
1
,
determine
the
scalar
pro
duct
−
→
w
·
−
→
t
.
E.8921
Prop
osition
:
for
an
y
v
ector
−
→
u
and
−
→
v
:
−
→
u
2
=
−
→
u
2
−
→
u
·
−
→
v
=
1
2
·
−
→
u
+
−
→
v
2
−
−
→
u
2
−
−
→
v
2
−
→
u
·
−
→
v
=
1
2
·
−
→
u
2
+
−
→
v
2
−
−
→
u
−
−
→
v
2
−
→
u
·
−
→
v
=
1
4
·
−
→
u
+
−
→
v
2
−
−
→
u
−
−
→
v
2
Consider
the
parallelepip
ed
AB
C
D
E
F
GH
sho
wn
b
elo
w
:
The
space
is
giv
en
the
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
sho
wn
ab
o
v
e.
W
e
then
ha
v
e
the
measuremen
ts
:
AB
=
10
;
AD
=
5
;
AE
=
4
1
Determine
the
v
alues
of
−
→
AC
,
−
−
→
C
G
et
−
→
AC
·
−
−
→
C
G
.
2
Deduce
the
measure
of
diagonal
[
AG
]
.
8.
Sc
alar
pr
o
duct
and
ortho
gonality
E.2778
In
space
with
an
orthonormal
refer-
ence
frame,
consider
the
follo
wing
three
p
oin
ts
:
A
6
;
5
;
1
;
B
−
4
;
2
;
−
4
;
C
4
;
7
;
2
Sho
w
that
the
triangle
AB
C
is
a
righ
t-angled
triangle
in
C
.
E.8893
In
space,
consider
the
cub
e
AB
C
D
E
F
GH
sho
wn
b
elo
w
and
where
the
p
oin
ts
I
,
J
,
K
are
the
resp
ectiv
e
middles
of
the
segmen
ts
[
AB
]
,
[
D
G
]
,
[
E
H
]
.
Consider
the
space
pro
vided
with
the
reference
frame
D
;
−
−
→
D
A
;
−
−
→
D
C
;
−
−
→
D
H
.
Sho
w
that
the
straigh
t
lines
(
I
J
)
and
(
C
K
)
are
orthogonal.
E.8920
Consider
the
parallelepip
ed
AB
C
D
E
F
GH
sho
wn
b
elo
w
where
:
AB
=
a
with
a
∈
R
∗
+
;
AD
=
1
;
AE
=
1
1
Justify
that
the
straigh
t
lines
(
H
B
)
and
(
AG
)
are
copla-
nar.
2
Determine
the
v
alue
of
a
so
that
the
straigh
t
lines
(
H
B
)
and
(
AG
)
are
p
erp
endicular.
9.
Determine
the
me
asur
e
of
an
angle
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8921
sacados/8921
A
B
C
D
E
F
G
H
−
i
−
k
−
j
chapExoCorrec/2778
sacados/2778
chapExoCorrec/8893
sacados/8893
I
J
K
A
B
C
D
E
F
G
H
chapExoCorrec/8920
sacados/8920
A
B
C
D
E
F
G
H
A
B
C
D
E
F
G
H
M
E.4082
Space
is
referred
to
a
refer-
ence
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
orthonormal
direct.
Consider
the
p
oin
ts
:
A
−
2
;
0
;
1
;
B
1
;
2
;
−
1
;
C
−
2
;
2
;
2
1
Calculate
the
scalar
pro
duct
−
−
→
AB
·
−
→
AC
then
the
lengths
AB
and
AC
.
2
Deduce
an
appro
ximate
v
alue,
rounded
to
the
nearest
degree,
of
the
angle
∠
B
AC
.
3
Justify
that
the
p
oin
ts
A
,
B
and
C
are
not
aligned.
E.4310
In
an
orthonormal
space
frame,
consider
the
three
p
oin
ts
:
E
2
;
1
;
−
3
;
F
1
;
−
1
;
2
;
G
−
1
;
3
;
1
Indicate
whether
the
follo
wing
statemen
t
is
true
or
false,
jus-
tifying
y
our
answ
er
:
a
measure
in
degrees
of
the
geometric
angle
∠
F
E
G
,
rounded
to
the
degree,
is
50
o
.
E.6967
Consider
a
cuble
AB
C
D
E
F
GH
whose
ca
v
alier
p
ersp
ectiv
e
represen
tation
is
giv
en
b
elo
w
:
The
edges
are
of
length
1
.
Space
is
referred
to
the
orthonor-
mal
reference
frame
D
;
−
−
→
D
A
;
−
−
→
D
C
;
−
−
→
D
H
T
o
an
y
real
x
of
the
in
terv
al
0
;
1
,
w
e
asso
ciate
the
p
oin
t
M
of
the
segmen
t
[
D
F
]
suc
h
that
:
−
−
→
D
M
=
x
·
−
−
→
D
F
W
e
are
in
terested
in
the
ev
olution
of
the
measure
„
in
radian
of
the
angle
∠
E
M
B
when
the
p
oin
t
M
tra
v
erses
the
segmen
t
[
D
F
]
.
W
e
ha
v
e
:
0
„
ı
1
What
is
„
if
the
p
oin
t
M
is
confused
with
the
p
oin
t
D
?
with
the
p
oin
t
F
?
2
Assume
that
the
p
oin
t
M
has
co
ordinates
M
(
x
;
x
;
x
)
.
Sho
w
that
:
cos
(
„
)
=
3
·
x
2
−
4
·
x
+1
3
·
x
2
−
4
·
x
+2
(F
or
this,
we
c
an
lo
ok
at
the
sc
alar
pr
o
duct
of
the
ve
ctors
−
−
→
M
E
and
−
−
→
M
B
)
E.10377
In
space
with
an
orthonormal
ref-
erence
frame,
consider
the
three
p
oin
ts
:
A
5
;
−
3
;
1
;
B
1
;
2
;
3
;
C
4
;
−
3
;
−
1
1
Demonstrate
that
the
triangle
AB
C
is
righ
t-angled
at
A
.
2
Calculate
the
scalar
pro
duct
−
−
→
B
A
·
−
−
→
B
C
and
then
the
lengths
AB
and
B
C
.
3
Deduce
the
measure
in
degrees
of
the
angle
∠
AB
C
rounded
to
the
degree.
E.10378
In
space
with
an
orthonormal
ref-
erence
frame,
consider
the
three
p
oin
ts
:
A
−
2
;
4
;
0
;
B
8
;
−
5
;
1
;
C
−
3
;
3
;
1
1
Demonstrate
that
the
triangle
AB
C
is
righ
t-angled
at
A
.
2
Calculate
the
scalar
pro
duct
−
−
→
B
A
·
−
−
→
B
C
and
then
the
lengths
AB
and
B
C
.
3
Deduce
the
measure
in
degrees
of
the
angle
∠
AB
C
rounded
to
the
degree.
E.10379
In
space
with
an
orthonormal
ref-
erence
frame,
consider
the
three
p
oin
ts
:
A
1
;
−
3
;
−
5
;
B
0
;
2
;
−
2
;
C
1
;
−
3
;
−
5
1
Demonstrate
that
the
triangle
AB
C
is
righ
t-angled
at
A
.
2
Calculate
the
scalar
pro
duct
−
−
→
B
A
·
−
−
→
B
C
and
then
the
lengths
AB
and
B
C
.
3
Deduce
the
measure
in
degrees
of
the
angle
∠
AB
C
rounded
to
the
degree.
10.
V
e
ctors
normal
to
a
plane
E.5411
Definition
:
let
P
b
e
a
plane
of
space
admitting
the
v
ectors
−
→
u
and
−
→
v
non-colinear
as
director
v
ectors.
A
v
ector
−
→
n
is
said
to
b
e
normal
to
the
plane
P
if
it
is
orthogonal
to
eac
h
of
the
v
ectors
−
→
u
and
−
→
v
.
In
space
with
a
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
orthonormal,
consider
the
three
p
oin
ts
A
,
B
and
C
with
co
ordinates
:
A
0
;
1
;
−
1
;
B
1
;
−
1
;
−
8
;
C
−
1
;
0
;
0
1
Determine
the
scalar
pro
ducts
−
−
→
AB
·
−
→
u
and
−
→
AC
·
−
→
u
.
2
What
can
b
e
said
ab
out
the
v
ector
−
→
u
3
;
−
2
;
1
relativ
e
to
the
plane
(
AB
C
)
.
E.5412
In
space
pro
vided
with
a
O
;
−
→
i
;
−
→
j
;
−
→
k
orthonormal,
consider
the
three
p
oin
ts
A
,
B
and
C
with
co
ordinates
:
A
1
;
0
;
0
;
B
1
;
1
;
1
;
C
7
;
2
;
−
1
Sho
w
that
the
v
ector
−
→
u
−
1
;
2
;
−
2
is
a
normal
v
ector
to
the
plane
(
AB
C
)
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/4082
sacados/4082
Extrait de Nouvelle-Caledonie Mars 2011
chapExoCorrec/4310
sacados/4310
chapExoCorrec/6967
sacados/6967
Extrait Liban Juin 2017
A
B
C
D
E
F
G
H
M
sacados/10377
sacados/10378
sacados/10379
chapExoCorrec/5411
sacados/5411
chapExoCorrec/5412
sacados/5412
A
B
C
D
E
F
G
H
I
J
K
A
B
C
D
E
F
G
H
I
J
E.8900
AB
C
D
E
F
GH
denotes
a
cub
e
with
side
1
.
The
p
oin
t
I
is
the
middle
of
the
segmen
t
[
B
F
]
.
P
oin
t
J
is
the
middle
of
segmen
t
[
B
C
]
.
P
oin
t
K
is
the
middle
of
segmen
t
[
C
D
]
.
Ab
o
v
e
is
sho
wn
the
(
I
J
K
)
plane.
Space
is
referred
to
the
reference
frame
A
;
−
−
→
AB
;
−
−
→
AD
;
−
→
AE
.
1
Giv
e
the
co
ordinates
A
,
G
,
I
,
J
and
K
in
this
frame
of
reference.
2
Sho
w
that
the
v
ector
−
→
AG
is
normal
to
the
plane
(
I
J
K
)
.
11.
Determine
a
normal
ve
ctor
to
a
plane
E.8919
In
the
plane
pro
vided
with
a
O
;
−
→
i
;
−
→
j
;
−
→
k
orthonormal,
consider
a
plane
(
P
)
admitting
the
v
ectors
−
→
u
2
;
1
;
1
and
−
→
v
2
;
−
1
;
−
2
non-colinear
as
di-
rector
v
ectors.
Let
−
→
n
(
x
;
y
;
z
)
b
e
a
normal
v
ector
to
the
(
P
)
plane.
1
Sho
w
that
the
co
ordinates
of
the
v
ector
−
→
n
v
erify
the
system
:
2
x
+
y
+
z
=
0
2
x
−
y
−
2
z
=
0
2
W
e
note
−
→
n
the
v
ector
normal
to
the
plane
(
P
)
ha
ving
1
for
dimension
and
w
e
note
its
co
ordinates
:
−
→
n
x
;
y
;
1
a
Determine
the
co
ordinates
of
the
v
ector
−
→
n
.
b
Prop
ose
a
v
ector
−
→
n
normal
to
the
plane
(
P
)
with
in
teger
co
ordinates.
E.5415
In
space
pro
vided
with
a
O
;
−
→
i
;
−
→
j
;
−
→
k
,
consider
the
three
p
oin
ts
A
,
B
and
C
with
co
ordinates
:
A
2
;
−
1
;
−
1
;
B
−
1
;
3
;
1
;
C
1
;
1
;
−
1
1
Sho
w
that
the
p
oin
ts
A
,
B
and
C
define
a
plane.
2
Determine
a
non-zero
−
→
u
v
ector
with
in
teger
co
ordinates
and
orthogonal
to
the
(
AB
C
)
plane.
E.8901
Space
is
referred
to
the
orthonormal
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
.
F
or
eac
h
question,
determine
the
co
ordinates
of
a
v
ector
−
→
n
non-zero
and
orthogonal
to
the
t
w
o
v
ectors
−
→
u
and
−
→
v
:
−
→
u
5
;
0
;
1
et
−
→
v
−
1
;
1
;
2
E.8902
Consider
the
rectangular
paral-
lelepip
ed
AB
C
D
E
F
GH
sho
wn
b
elo
w
:
P
oin
ts
I
and
J
are
the
resp
ectiv
e
middles
of
segmen
ts
[
AE
]
and
[
AB
]
.
The
middles
of
the
v
arious
edges
are
sho
wn
in
the
figure.
1
Sho
w
the
section
of
the
(
D
I
F
)
plane
and
the
paral-
lelepip
ed.
Justify
y
our
construction.
2
The
plane
is
giv
en
the
reference
frame
A
;
−
→
AJ
;
−
−
→
AD
;
−
→
AE
orthonormal.
Determine
a
v
ector
−
→
n
,
with
in
teger
co
or-
dinates,
normal
to
the
plane
(
D
I
F
)
.
12.
Ortho
gonal
pr
oje
ction
on
a
plane
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8900
sacados/8900
A
B
C
D
E
F
G
H
I
J
K
chapExoCorrec/8919
sacados/8919
chapExoCorrec/5415
sacados/5415
chapExoCorrec/8901
sacados/8901
chapExoCorrec/8902
sacados/8902
A
B
C
D
E
F
G
H
I
J
A
B
C
D
E
F
G
H
I
H
G
F
E
D
C
B
A
I
J
E.8905
Definition
-
prop
osition
:
In
m
uni
space,
consider
a
p
oin
t
A
and
a
plane
(
P
)
.
W
e
call
the
orthogonal
pro
jected
of
the
p
oin
t
A
on
to
the
plane
(
P
)
,
the
single
p
oin
t
M
in
tersection
of
the
plane
P
with
the
straigh
t
line
passing
through
the
p
oin
t
A
and
orthogonal
to
the
plane
P
Corollary
:
in
space,
consider
a
p
oin
t
A
and
a
plane
(
P
)
.
The
pro
jected
H
of
the
p
oin
t
A
on
to
the
plane
(
P
)
is
the
unique
p
oin
t
of
the
p
oin
t
(
P
)
suc
h
that
the
line
(
AH
)
is
orthogonal
to
the
plane
(
P
)
.
AB
C
D
E
F
GH
is
the
cub
e
with
edge
1
sho
wn
on
the
attac
hed
sheet
to
b
e
completed
and
returned
with
the
cop
y
.
Space
is
referred
to
the
orthonormal
reference
frame
A
;
−
−
→
AB
;
−
−
→
AD
;
−
→
AE
.
Consider
the
p
oin
t
I
with
co
ordinates
I
1
3
;
1
3
;
1
3
.
1
a
Establish
equalit
y
:
1
3
·
−
−
→
D
B
+
1
3
·
−
−
→
D
E
=
−
→
D
I
b
Sho
w
that
the
p
oin
t
I
b
elongs
to
the
plane
(
B
D
E
)
.
2
Sho
w
that
p
oin
t
I
is
the
pro
jected
p
oin
t
of
p
oin
t
A
on
to
plane
(
B
D
E
)
.
E.8918
In
the
plane
pro
vided
with
a
O
;
−
→
i
;
−
→
j
;
−
→
k
orthonormal,
consider
the
p
oin
t
M
5
;
2
;
−
1
and
the
plane
(
P
)
passing
through
the
p
oin
t
A
5
;
−
6
;
−
6
and
admitting
the
v
ector
−
→
n
2
;
1
;
2
as
normal
v
ector.
Sho
w
that
the
p
oin
t
H
1
;
0
;
−
5
is
the
orthogonal
pro
ject
of
the
p
oin
t
M
on
to
the
plane
(
P
)
.
E.8909
Consider
the
cub
e
AB
C
D
E
F
GH
,
with
edge
length
1
,
and
note
I
and
J
the
mid-
dles
of
the
[
AB
]
and
[
C
G
]
edges.
On
utilisera
le
rep
ère
A
;
−
−
→
AB
;
−
−
→
AD
;
−
→
AE
W
e
note
M
the
p
oin
t
with
co
ordinates
:
M
1
2
;
2
5
;
4
5
1
Sho
w
that
the
p
oin
t
M
is
the
pro
jected
p
oin
t
of
the
p
oin
t
I
on
the
plane
(
E
F
J
)
2
Sho
w
that
the
v
olume
of
the
tetrahedron
E
F
I
J
is
equal
to
1
6
E.10380
In
space
pro
vided
with
an
orthonormal
reference
frame,
consider
e
p
oin
t
A
1
;
2
;
3
and
the
plane
P
admitting
as
equation
:
P
:
2
·
x
−
2
·
y
−
4
·
z
+
2
=
0
Sho
w
that
the
p
oin
t
H
2
;
1
;
1
is
the
orthogonal
pro
ject
of
the
p
oin
t
A
on
to
the
plane
P
.
E.10381
In
space
pro
vided
with
an
orthonormal
reference
frame,
consider
e
p
oin
t
A
2
;
−
3
;
2
and
the
plane
P
admitting
as
equation
:
P
:
−
6
·
x
+
4
·
y
+
z
−
5
=
0
Sho
w
that
the
p
oin
t
H
−
1
;
−
1
;
3
is
the
orthogonal
pro
ject
of
the
p
oin
t
A
on
to
the
plane
P
.
E.10382
In
space
pro
vided
with
an
orthonormal
reference
frame,
consider
e
p
oin
t
A
1
;
2
;
2
and
the
plane
P
admitting
as
equation
:
P
:
x
+
2
·
y
+
3
·
z
−
5
=
0
Sho
w
that
the
p
oin
t
H
4
7
;
8
7
;
5
7
is
the
orthogonal
pro
ject
of
the
p
oin
t
A
on
to
the
plane
P
.
E.10383
In
space
pro
vided
with
an
orthonormal
reference
frame,
consider
the
p
oin
t
A
3
;
3
;
3
and
the
plane
P
admitting
as
standard
form
:
2
·
x
−
y
+
z
+
1
=
0
Determine
the
co
ordinates
of
the
p
oin
t
H
pro
jected
orthogo-
nally
from
the
p
oin
t
A
on
the
plane
P
.
E.10384
In
space
with
an
orthonormal
reference
frame,
consider
the
p
oin
t
A
8
3
;
4
3
;
−
8
3
and
the
plane
P
with
standard
form
:
x
+
2
·
y
−
z
+
2
=
0
Determine
the
co
ordinates
of
the
p
oin
t
H
pro
jected
orthogo-
nally
from
the
p
oin
t
A
on
the
plane
P
.
E.10385
In
space
pro
vided
with
an
orthonormal
reference
frame,
consider
the
p
oin
t
A
2
;
−
3
;
−
3
and
the
plane
P
admitting
as
standard
form
:
−
2
·
x
+
y
+
4
·
z
−
2
=
0
Determine
the
co
ordinates
of
the
p
oin
t
H
pro
jected
orthogo-
nally
from
the
p
oin
t
A
on
to
the
plane
P
.
E.10386
to
do
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8905
sacados/8905
A
B
C
D
E
F
G
H
I
chapExoCorrec/8918
sacados/8918
chapExoCorrec/8909
sacados/8909
H
G
F
E
D
C
B
A
I
J
sacados/10380
chapExoCorrec/10381
sacados/10381
sacados/10382
sacados/10383
sacados/10384
sacados/10385
sacados/10386
fichierPlus/10386/
A
B
C
D
E
F
G
H
I
I
H
H
G
F
E
D
C
B
A
13.
Distanc
e
to
a
plane
E.8910
Prop
osition
:
in
m
uni
space,
consider
a
p
oin
t
A
and
a
plane
(
P
)
.
Let
H
b
e
the
orthogonal
pro
ject
of
the
p
oin
t
A
on
to
the
plane
(
P
)
.
The
distance
AH
has
the
v
alue
:
AH
=
⏐
⏐
−
−
→
AB
·
−
→
n
⏐
⏐
−
→
n
Definition
:
in
space,
consider
a
p
oin
t
A
and
a
plane
(
P
)
,
w
e
call
distance
from
p
oin
t
A
to
plane
(
P
)
,
distance
from
p
oin
t
AH
where
p
oin
t
H
is
the
orthogonal
pro
ject
of
p
oin
t
A
on
to
plane
(
P
)
In
space
with
a
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
l
orthonormal,
consider
the
four
p
oin
ts
:
A
2
;
2
;
0
;
B
4
;
−
4
;
1
;
C
5
;
−
2
;
−
1
;
M
6
;
4
;
4
1
Sho
w
that
p
oin
ts
A
,
B
,
C
are
non-aligned.
2
Sho
w
that
the
v
ector
−
→
n
2
;
1
;
2
is
a
normal
v
ector
to
the
plane
(
AB
C
)
.
3
Determine
the
distance
from
p
oin
t
M
to
plane
(
AB
C
)
.
E.8906
Consider
a
cub
e
AB
C
D
E
F
GH
,
with
edge
length
1
.
Note
I
the
p
oin
t
of
in
tersection
of
the
line
(
E
C
)
and
the
plane
(
AF
H
)
.
In
space,
consider
the
cub
e
AB
C
D
E
F
GH
sho
wn
b
elo
w
and
use
the
reference
frame
D
;
−
−
→
D
A
;
−
−
→
D
C
;
−
−
→
D
H
orthonormé.
Let
I
b
e
the
orthogonal
pro
ject
of
the
p
oin
t
E
on
to
the
plane
(
AF
H
)
.
1
Justify
that
the
line
(
E
C
)
is
orthogonal
to
the
plane
(
AH
F
)
.
2
V
erify
that
the
distance
from
p
oin
t
E
to
plane
(
AF
H
)
is
equal
to
3
3
.
E.8911
Prop
osition
:
The
orthogonal
pro
ject
H
of
a
p
oin
t
A
on
a
plane
P
is
the
single
p
oin
t
of
the
plane
P
closest
to
A
Consider
the
cub
e
AB
C
D
E
F
GH
sho
wn
b
elo
w,
and
space
is
referred
to
the
orthonormal
reference
frame
A
;
−
−
→
AB
;
−
−
→
AD
;
−
→
AE
.
Note
I
the
p
oin
t
with
co
ordinates
1
3
;
1
;
1
and
the
section
through
the
plane
(
F
H
I
)
is
sho
wn
shaded.
1
Demonstrate
that
the
v
ector
−
→
n
with
co
ordinates
3
;
−
3
;
2
is
normal
to
the
plane
(
AC
I
)
2
Determine
the
distance
of
p
oin
t
F
from
the
plane.
3
Let
H
b
e
the
orthogonal
pro
ject
of
the
p
oin
t
F
on
to
the
plane
(
AC
I
)
.
Sho
w
that
the
p
oin
t
H
has
co
ordinates
:
H
7
22
;
15
22
;
12
22
14.
Ortho
gonal
pr
oje
ction
on
a
str
aight
line
E.8915
Definition
-
prop
osition
:
In
space,
consider
a
p
oin
t
A
and
a
straigh
t
line
(
d
)
with
director
v
ector
−
→
u
.
W
e
call
the
orthogonal
pro
jected
of
the
p
oin
t
A
on
to
the
line
(
d
)
,
the
p
oin
t
of
in
tersection
of
the
line
(
d
)
with
the
plane
passing
through
the
p
oin
t
A
and
admitting
−
→
u
as
normal
v
ector.
Corollary
:
in
space,
consider
a
straigh
t
line
(
d
)
and
a
p
oin
t
A
not
b
elonging
to
(
d
)
.
The
orthogonal
pro
ject
of
the
p
oin
t
A
on
to
the
line
(
d
)
is
the
only
p
oin
t
M
b
elonging
to
(
d
)
and
suc
h
that
the
v
ector
−
−
→
AM
is
orthogonal
to
an
y
directing
v
ector
of
the
line
(
d
)
.
In
space
pro
vided
with
a
reference
frame
O
;
;
−
→
i
;
−
→
j
or-
thonormal,
consider
the
three
p
oin
ts
:
A
7
;
3
;
0
;
B
6
;
−
1
;
−
8
;
C
−
4
;
4
;
2
Consider
the
p
oin
t
H
2
;
1
;
−
4
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8910
sacados/8910
chapExoCorrec/8906
sacados/8906
A
B
C
D
E
F
G
H
I
chapExoCorrec/8911
sacados/8911
I
H
H
G
F
E
D
C
B
A
chapExoCorrec/8915
sacados/8915
A
B
C
D
E
F
G
H
I
J
A
B
C
D
S
I
1
Sho
w
that
the
p
oin
t
H
b
elongs
to
the
line
(
B
C
)
.
2
Establish
that
H
is
the
orthogonal
pro
ject
of
the
p
oin
t
A
on
to
the
line
(
B
C
)
.
E.8903
AB
C
D
E
F
GH
denotes
a
cub
e
with
side
1
,
the
p
oin
t
I
is
the
midp
oin
t
of
the
segmen
t
[
B
F
]
and
the
p
oin
t
J
is
the
midp
oin
t
of
the
segmen
t
[
AG
]
.
Space
is
referred
to
the
reference
frame
A
;
−
−
→
AB
;
−
−
→
AD
;
−
→
AE
.
1
Establish
that
the
p
oin
t
J
is
the
pro
jected
p
oin
t
of
the
p
oin
t
I
on
to
the
line
(
AG
)
.
2
Determine
the
distance
I
J
.
E.8916
In
space,
consider
the
AB
C
D
S
p
yra-
mid
with
a
square
base
and
side
faces
that
are
all
equilateral
triangles.
Let
I
b
e
the
midp
oin
t
of
segmen
t
[
AB
]
.
W
e
m
uni
the
space
of
the
A
;
−
−
→
AB
;
−
−
→
AD
;
−
→
k
direct
orthonor-
mal
reference
frame.
1
Justify
that
the
v
ector
−
→
C
S
has
co
ordinates
:
−
→
C
S
−
1
2
;
−
1
2
;
2
2
Note
H
(
x
;
y
;
z
)
the
orthogonal
pro
ject
of
the
p
oin
t
A
on
to
the
line
(
C
S
)
.
2
Justify
that
there
exists
a
real
k
allo
wing
to
write
the
co
ordinates
of
the
p
oin
t
H
:
H
−
1
2
·
k
+1
;
−
1
2
·
k
+1
;
2
2
·
k
3
Deduce
the
co
ordinates
of
the
p
oin
t
H
.
15.
Distanc
e
fr
om
a
str
aight
line
E.8912
Prop
osition
:
in
space
pro
vided
with
an
orthonormal
ref-
erence
frame,
consider
a
straigh
t
line
(
d
)
passing
through
a
p
oin
t
A
and
admitting
−
→
u
as
its
directing
v
ector
and
B
another
p
oin
t
of
the
plane.
Noting
H
the
pro
jected
p
oin
t
B
on
the
line
(
d
)
,
w
e
ha
v
e
:
B
H
=
−
−
→
B
A
−
−
−
→
B
A
·
−
→
u
−
→
u
2
−
→
u
Definition
:
in
space
pro
vided
with
an
orthonormal
refer-
ence
frame,
consider
a
straigh
t
line
(
d
)
and
a
p
oin
t
B
in
space.
W
e
call
distance
from
the
p
oin
t
B
to
the
line
(
d
)
,
distance
B
H
where
H
is
the
pro
jected
p
oin
t
B
on
the
line
(
d
)
.
Space
is
referred
to
a
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
or-
thonormal
and
consider
the
p
oin
ts
A
,
B
and
C
whose
co-
ordinates
are
:
A
5
;
0
;
1
;
B
−
5
;
2
;
8
;
C
7
;
−
4
;
−
4
Establish
that
the
p
oin
t
A
is
at
a
distance
of
3
from
the
line
(
B
C
)
.
E.8914
Prop
osition
:
the
orthogonal
pro
jected
H
of
the
p
oin
t
A
on
the
line
(
d
)
is
the
only
p
oin
t
on
the
line
(
d
)
closest
to
the
p
oin
t
A
.
Space
is
referred
to
a
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
or-
thonormal
and
consider
the
p
oin
ts
A
,
B
and
C
whose
co-
ordinates
are
:
A
3
;
4
;
−
7
;
B
−
5
;
−
1
;
−
6
;
C
−
1
;
−
3
;
−
2
W
e
denote
b
y
H
the
orthogonal
pro
ject
of
the
p
oin
t
A
on
the
straigh
t
line
(
B
C
)
.
1
Determine
the
distance
from
p
oin
t
A
to
line
(
B
C
)
.
2
Establish
that
the
p
oin
t
H
has
co
ordinates
:
H
−
3
;
−
2
;
−
4
h
ttps:/
/c
hingmath.fr
sacados/8903
A
B
C
D
E
F
G
H
I
J
chapExoCorrec/8916
sacados/8916
A
B
C
D
S
I
chapExoCorrec/8912
sacados/8912
chapExoCorrec/8914
sacados/8914
A
B
C
D
E
F
G
H
I
H
G
F
E
D
C
B
A
I
I
R
H
G
F
E
D
C
B
A
E.8907
AB
C
D
E
F
GH
denotes
a
cub
e
with
side
1
,
the
p
oin
t
I
is
the
midp
oin
t
of
the
segmen
t
[
B
F
]
.
Space
is
referred
to
the
reference
frame
A
;
−
−
→
AB
;
−
−
→
AD
;
−
→
AE
.
Determine
the
distance
from
p
oin
t
I
to
line
(
AG
)
.
E.8904
Consider
the
cub
e
AB
C
D
E
F
GH
sho
wn
on
the
attac
hed
sheet.
Throughout
the
exer-
cise,
space
is
referred
to
the
orthonormal
reference
frame
A
;
−
−
→
AB
;
−
−
→
AD
;
−
→
AE
.
Let
I
b
e
the
p
oin
t
with
co
ordinates
1
3
;
1
;
1
and
R
the
orthogonal
pro
ject
of
I
on
to
the
line
(
AC
)
.
1
Establish
that
:
I
R
=
11
3
.
2
The
p
oin
t
R
b
elonging
to
the
line
(
AC
)
,
there
exists
a
real
t
suc
h
that
:
−
→
AR
=
t
·
−
→
AC
a
Express
the
co
ordinates
of
p
oin
t
R
as
a
function
of
t
.
b
Using
question
1
,
determine
the
co
ordinates
of
p
oin
t
R
.
E.4100
Space
is
referred
to
an
orthonor-
mal
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
.
The
p
oin
ts
A
,
B
and
C
ha
v
e
the
resp
ectiv
e
co
ordinates
:
A
1
;
−
2
;
4
;
B
−
2
;
−
6
;
5
;
C
−
4
;
0
;
−
3
W
e
denote
b
y
H
the
orthogonal
pro
ject
of
the
p
oin
t
O
on
the
line
(
B
C
)
.
Let
t
b
e
the
real
suc
h
that
:
−
−
→
B
H
=
t
·
−
−
→
B
C
1
Sho
w
that
:
t
=
−
−
→
B
O
·
−
−
→
B
C
−
−
→
B
C
2
2
Deduce
the
real
t
and
the
co
ordinates
of
the
p
oin
t
H
.
16.
Sc
alar
pr
o
duct
and
ve
ctor
r
elation
E.4094
Consider
a
tetrahedron
AB
C
D
and
note
H
the
orthogonal
pro
ject
of
the
p
oin
t
A
on
to
the
plane
(
B
C
D
)
.
Sho
w
that,
if
the
heigh
ts
of
the
tetrahedron
AB
C
D
arising
from
the
p
oin
ts
A
and
B
are
concurren
t,
then
the
line
(
B
H
)
is
a
heigh
t
of
the
triangle
B
C
D
.
17.
Unclassifie
d
financial
ye
ars
E.4247
Consider
a
cub
e
AB
C
D
E
F
GH
of
edge
length
3
W
e
c
ho
ose
the
orthonormal
reference
frame
D
;
−
→
i
;
−
→
j
;
−
→
k
suc
h
that
:
−
→
i
=
1
3
·
−
−
→
D
A
;
−
→
j
=
1
3
·
−
−
→
D
C
;
−
→
k
=
1
3
·
−
−
→
D
H
h
ttps:/
/c
hingmath.fr
chapExoCorrec/8907
sacados/8907
A
B
C
D
E
F
G
H
I
chapExoCorrec/8904
sacados/8904
H
G
F
E
D
C
B
A
I
I
R
sacados/4100
chapExoCorrec/4094
sacados/4094
sacados/4247
Extrait de Polynesie Septembre 2007
H
G
F
E
D
C
B
A
1
Determine
the
co
ordinates
of
the
p
oin
t
L
barycen
ter
of
the
system
(
C
;
2
)
;
(
E
;
1
)
.
The
v
ectors
−
→
AE
and
−
→
D
L
are
assumed
to
ha
v
e
the
follo
wing
co
ordinates
:
−
→
AE
0
;
0
;
3
;
−
→
D
L
1
;
2
;
1
Let
(
a
;
b
)
b
e
a
pair
of
reals.
Let
M
b
e
the
p
oin
t
on
the
line
(
AE
)
suc
h
that
:
−
−
→
AM
=
a
·
−
→
AE
and
N
the
p
oin
t
on
the
line
(
D
L
)
suc
h
that
:
−
−
→
D
N
=
b
·
−
→
D
L
2
Sho
w
that
the
v
ector
−
−
→
M
N
is
orthogonal
to
the
v
ectors
−
→
AE
and
−
→
D
L
if
and
only
if
the
pair
(
a
;
b
)
v
érifie
the
sys-
tem
:
−
a
+
2
b
=
1
3
a
−
b
=
0
3
Deduce
that
there
is
a
single
p
oin
t
M
0
of
(
AE
)
and
a
sin-
gle
p
oin
t
N
0
of
(
D
L
)
suc
h
that
the
straigh
t
line
(
M
0
N
0
)
is
orthogonal
to
the
straigh
t
lines
(
AE
)
and
(
D
L
)
.
4
Determine
the
co
ordinates
of
the
p
oin
ts
M
0
and
N
0
then
calculate
the
distance
M
0
N
0
.
E.4034
Space
is
referred
to
the
orthonormal
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
.
The
sphere
of
cen
ter
A
1
;
1
;
1
and
radius
10
is
tangen
t
to
the
plane
P
of
equation
x
+
y
+
z
=
0
.
E.4110
Space
is
referred
to
the
orthonormal
reference
frame
O
;
−
→
i
;
−
→
j
;
−
→
k
.
Consider
a
plane
(
P
)
admitting
the
v
ectors
−
→
u
and
−
→
v
nonco-
linear
v
ectors
and
whose
co
ordinates
are
:
−
→
u
2
;
−
1
;
3
et
−
→
v
−
1
;
1
;
1
Note
−
→
n
(
x
;
y
;
z
)
a
normal
v
ector
to
the
plane
(
P
)
1
Justify
that
the
co
ordinates
of
the
v
ector
−
→
n
are
solutions
of
the
system
of
equations
:
2
x
−
y
+
3
z
=
0
−
x
+
y
+
z
=
0
2
a
Cho
osing
z
=
1
,
determine
the
co
ordinates
of
the
sin-
gle
v
ector
−
→
n
1
normal
to
the
plane
and
whose
coast
is
1
.
b
Determine
the
co
ordinates
of
another
v
ector
−
→
n
normal
to
the
plane
(
P
)
.
c
What
can
w
e
sa
y
ab
out
the
v
ectors
−
→
n
1
and
−
→
n
?
E.4130
Consider
the
space
pro
vided
with
a
O
;
−
→
i
;
−
→
j
;
−
→
j
orthonormal
reference
frame.
Consider
the
line
(
d
)
passing
through
the
p
oin
t
A
1
;
−
2
;
3
and
admitting
the
v
ector
−
→
u
−
2
;
0
;
1
as
director
v
ector.
Let
M
b
e
the
p
oin
t
in
space
with
co
ordinates
1
;
−
1
;
13
,
determine
the
co
ordinates
of
the
pro
jected
H
of
the
p
oin
t
M
on
the
line
(
d
)
.
E.8908
Consider
the
space
pro
vided
with
a
O
;
−
→
i
;
−
→
j
;
−
→
j
orthonormal
reference
frame.
Let
(
P
)
b
e
the
plane
admitting
as
standard
form
:
3
·
x
−
y
+
2
·
z
=
0
Consider
the
p
oin
t
N
15
;
1
;
6
.
Determine
the
co
ordinates
of
the
p
oin
t
I
pro
jected
orthogonally
from
the
p
oin
t
N
in
the
plane
(
P
)
.
h
ttps:/
/c
hingmath.fr
sacados/4034
Extrait Antilles-Guyane Septembre 2010
chapExoCorrec/4110
sacados/4110
chapExoCorrec/4130
sacados/4130
chapExoCorrec/8908
sacados/8908
ChingMath
by ChingLab
©
Middle School
Grade 6
Grade 7
Grade 8
Grade 9
High School
Grade 10
Grade 11
Grade 12
Grade 12 opt. Complémentaire
Grade 12 opt. Experte
Various
Contact me
F.A.Q
Other website
ChingWeb