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Byju's Answer
Standard XII
Mathematics
Shortest Distance between Two Skew Lines
Find the shor...
Question
Find the shortest distance between two lines whose vector equations are
→
r
=
(
^
i
+
2
^
j
+
3
^
k
)
+
β
(
^
i
−
3
^
j
+
2
^
k
)
and
→
r
(
4
^
i
+
5
^
j
+
6
^
k
)
+
μ
(
2
^
i
+
3
^
j
+
^
k
)
.
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Solution
Line:
→
r
=
(
^
i
+
2
^
j
+
3
^
k
)
+
β
(
^
i
−
3
^
j
+
2
^
k
)
−
(
i
)
→
r
=
(
4
^
i
+
5
^
j
+
6
^
k
)
+
μ
(
2
^
i
+
3
^
j
+
^
k
)
−
(
i
i
)
→
a
=
^
i
+
2
^
j
+
3
^
k
,
→
b
=
4
^
i
+
5
^
j
+
6
^
k
are two points on Line
(
i
)
and
(
i
i
)
.
∴
−
−
→
A
B
=
3
^
i
+
3
^
j
+
3
^
k
Direction of shortest distance is perpendicular to both lines.
∴
→
n
=
∣
∣ ∣ ∣
∣
^
i
^
j
^
k
2
3
1
1
−
3
2
∣
∣ ∣ ∣
∣
=
9
^
i
−
3
^
j
+
9
^
k
∴
Perpendicular distance
=
−
−
→
A
B
.
→
n
∣
∣
→
n
∣
∣
=
(
3
^
i
+
3
^
j
+
3
^
k
)
.
(
9
^
i
−
3
^
j
+
9
^
k
)
√
9
2
+
9
2
+
1
2
=
3
√
19
u
n
i
t
s
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Q.
Find the shortest distance between the lines whose vector equations are