Chapter 11 Three Dimensional Geometry Worksheet Page 39
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41THREE DIMENSIONAL GEOMETRY
501
®
Shortest distance between two skew lines is the line segment perpendicular
to both the lines.
H
H
H
H
H
H
®
=
+ µ
=
+ λ
Shortest distance between
and
is
r
a
b
r
a
b
2
2
1
1
H
H
H
H
×
⋅
b
b
a
a
(
) (
–
)
H
H
1
2
2
1
×
|
b
b
|
1
2
−
−
−
x
x
y
y
z z
®
=
=
1
1
1
Shortest distance between the lines:
and
a
b
c
1
1
1
−
−
−
z z
x
x
y
y
=
2
2
2
=
is
a
b
c
2
2
2
−
−
−
x
x
y
y
z
z
2
1
2
1
2
1
a
b
c
1
1
1
a
b
c
2
2
2
−
+
−
+
−
2
2
2
(
b c
b c
)
(
c a
c a
)
(
a b
a b
)
1 2
2 1
1 2
2 1
1 2
2 1
H
H
H
H
H
H
®
=
+ λ
=
+ µ
Distance between parallel lines
and
is
r
a
b
r
a
b
1
2
H
H
H
×
−
b
(
a
a
)
H
2
1
| |
b
®
In the vector form, equation of a plane which is at a distance d from the
n ˆ
origin, and
is the unit vector normal to the plane through the origin is
H
⋅ =
ˆ
.
r n
d
®
Equation of a plane which is at a distance of d from the origin and the direction
cosines of the normal to the plane as l, m, n is lx + my + nz = d.
H
®
The equation of a plane through a point whose position vector is
and
a
KH
KH
H
H
−
=
perpendicular to the vector
is
.
N
(
r
a
) . N
0
®
Equation of a plane perpendicular to a given line with direction ratios A, B, C
and passing through a given point (x
, y
, z
) is
1
1
1
A (x – x
) + B (y – y
) + C (z – z
) = 0
1
1
1
®
Equation of a plane passing through three non collinear points (x
, y
, z
),
1
1
1
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