Chapter 11 Three Dimensional Geometry Worksheet Page 35
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41THREE DIMENSIONAL GEOMETRY
497
Adding third column to the first column, we get
−
−
+ −
−
b
a
b
a b c
a
d
α
α
α + δ
2
= 0
β
β
β + γ
Since the first and second columns are identical. Hence, the given two lines are
coplanar.
Example 30
Find the coordinates of the point where the line through the points
A (3, 4, 1) and B (5, 1, 6) crosses the XY-plane.
Solution
The vector equation of the line through the points A and B is
H
+
+
+ λ
−
+
−
+
−
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
=
3
i
4
j
k
[ (5 3)
i
(1 4)
j
( 6 1)
k
]
r
H
+
+
+ λ
−
+
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
i.e.
=
... (1)
3
i
4
j
k
( 2
i
3
j
5
k
)
r
Let P be the point where the line AB crosses the XY-plane. Then the position
ˆ +
ˆ
vector of the point P is of the form
i x
y
j
.
This point must satisfy the equation (1).
(Why ?)
+
+ λ
+
− λ
+
+ λ
ˆ
ˆ
ˆ
ˆ
ˆ
i.e.
=
x i
y j
(3 2 )
i
( 4 3 )
j
( 1 5 )
k
ˆ
ˆ ˆ
Equating the like coefficients of
, we have
i j
,
and
k
x = 3 + 2 λ
y = 4 – 3 λ
0 = 1 + 5 λ
Solving the above equations, we get
13
23
y =
and
x =
5
5
13
23
,
0 ,
Hence, the coordinates of the required point are
.
5
5
Miscellaneous Exercise on Chapter 11
1.
Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the
line determined by the points (3, 5, – 1), (4, 3, – 1).
2.
If l
, m
, n
and l
, m
, n
are the direction cosines of two mutually perpendicular
1
1
1
2
2
2
lines, show that the direction cosines of the line perpendicular to both of these
−
−
−
m n
m n
,
n l
n l
,
l m
l m
are
1
2
2
1
1 2
2 1
1
2
2
1
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