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Mathematics
The shortest distance between the lines vecr=(8+3λ) hati-(9+16λ) hatj+(10+7λ) hatk and vecr=15 hati+29 hatj+5 hatk+μ(3 hati-8 hatj-5 hatk) is
Q. The shortest distance between the lines
r
=
(
8
+
3
λ
)
i
^
−
(
9
+
16
λ
)
j
^
+
(
10
+
7
λ
)
k
^
and
r
=
15
i
^
+
29
j
^
+
5
k
^
+
μ
(
3
i
^
−
8
j
^
−
5
k
^
)
is
5916
190
Three Dimensional Geometry
Report Error
A
84
20%
B
14
30%
C
21
37%
D
16
13%
Solution:
Given, equation of lines are
r
=
(
8
+
3
λ
)
i
^
−
(
9
+
16
λ
)
j
^
+
(
10
+
7
λ
)
k
^
i.e.,
r
=
8
i
^
−
9
j
^
+
10
k
^
+
λ
(
3
i
^
−
16
j
^
+
7
k
^
)
and
r
=
15
i
^
+
29
j
^
+
5
k
^
+
μ
(
3
i
^
+
8
j
^
−
5
k
^
)
Here,
a
1
=
8
i
^
−
9
j
^
+
10
k
^
,
b
1
=
3
i
^
−
16
j
^
+
7
k
^
a
2
=
15
i
^
+
29
j
^
+
5
k
^
,
b
2
=
3
i
^
+
8
j
^
−
5
k
^
a
2
−
a
1
=
(
15
i
^
+
29
j
^
+
5
k
^
)
−
(
8
i
^
−
9
j
^
+
10
k
^
)
=
7
i
^
+
38
j
^
−
5
k
^
b
1
×
b
2
=
i
^
(
80
−
56
)
−
j
^
(
−
15
−
21
)
+
k
^
(
24
+
48
)
=
(
24
i
^
+
36
j
^
+
72
k
^
)
∴
Shortest distance,
d
=
∣
∣
∣
b
1
×
b
2
∣
(
a
2
−
a
1
)
⋅
(
b
1
×
b
2
)
∣
∣
=
∣
∣
(
24
)
2
+
(
36
)
2
+
(
72
)
2
(
7
i
^
+
38
j
^
−
5
k
^
)
⋅
(
24
i
^
+
36
j
^
+
72
k
^
)
∣
∣
=
∣
∣
576
+
1296
+
5184
168
+
1368
−
360
∣
∣
=
∣
∣
7056
1176
∣
∣
=
84
1176
=
14