Chapter 11 Three Dimensional Geometry Worksheet Page 40
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41502
MATHEMATICS
(x
, y
, z
) and (x
, y
, z
) is
2
2
2
3
3
3
−
−
−
x
x
y
y
z
z
1
1
1
−
−
−
x
x
y
y
z
z
2
1
2
1
2
1
= 0
−
−
−
x
x
y
y
z
z
3
1
3
1
3
1
®
Vector equation of a plane that contains three non collinear points having
H
H
H
H
H
H
H
H
H
−
−
×
−
=
position vectors
a
,
b
and
c
is
(
r
a
) . [ (
b
a
) (
c
a
) ]
0
®
Equation of a plane that cuts the coordinates axes at (a, 0, 0), (0, b, 0) and
(0, 0, c) is
x
y
z
+
+
=
1
a
b
c
®
Vector equation of a plane that passes through the intersection of
H H
H H
H H
H
⋅
=
⋅
=
⋅
+ λ
=
+ λ
, where λ is any
planes
is
r n
d
and
r n
d
r n
(
n
)
d
d
1
1
2
2
1
2
1
2
nonzero constant.
®
Vector equation of a plane that passes through the intersection of two given
planes A
x + B
y + C
z + D
= 0 and A
x + B
y + C
z + D
= 0
1
1
1
1
2
2
2
2
) + λ(A
is (A
x + B
y + C
z + D
x + B
y + C
z + D
) = 0.
1
1
1
1
2
2
2
2
H
H
H
H
H
H
®
=
+ λ
=
+ µ
Two planes
and
are coplanar if
r
a
b
r
a
b
1
1
2
2
H
H
H
H
−
⋅
×
= 0
(
a
a
) (
b
b
)
2
1
1
2
®
Two planes a
x + b
y + c
z + d
= 0 and a
x + b
y + c
z + d
= 0 are
1
1
1
1
2
2
2
2
−
−
−
x
x
y
y
z
z
2
1
2
1
2
1
a
b
c
coplanar if
1
1
1
= 0.
a
b
c
2
2
2
H H
®
In the vector form, if θ is the angle between the two planes,
⋅
=
and
r n
d
1
1
H H
⋅
H H
|
n n
|
⋅
=
, then θ = cos
1
2
H H
–1
r n
d
.
2
2
| | | |
n n
H
1
2
H
H
H
®
The angle φ between the line
= + λ
⋅ =
and the plane
ˆ
is
r
a
b
r n
d
H
⋅
ˆ
b n
φ =
H
sin
ˆ
| | | |
b n
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