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Byju's Answer
Standard XII
Mathematics
Differentiation to Solve Modified Sum of Binomial Coefficients
Show that the...
Question
Show that the lines
r
→
=
2
j
^
-
3
k
^
+
λ
i
^
+
2
j
^
+
3
k
^
and
r
→
=
2
i
^
+
6
j
^
+
3
k
^
+
μ
2
i
^
+
3
j
^
+
4
k
^
are coplanar. Also, find the equation of the plane containing them.
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Solution
We know that the lines
r
→
=
a
1
→
+
λ
b
1
→
and
r
→
=
a
2
→
+
μ
b
2
→
are coplanar if
a
1
→
.
b
1
→
×
b
2
→
=
a
2
→
.
b
1
→
×
b
2
→
and the equation of the plane containing them is
r
→
.
b
1
→
×
b
2
→
=
a
1
→
.
b
1
→
×
b
2
→
.
Here,
a
1
→
=
0
i
^
+
2
j
^
-
3
k
^
;
b
1
→
=
i
^
+
2
j
^
+
3
k
^
;
a
2
→
=
2
i
^
+
6
j
^
+
3
k
^
;
b
2
→
=
2
i
^
+
3
j
^
+
4
k
^
b
1
→
×
b
2
→
=
i
^
j
^
k
^
1
2
3
2
3
4
=
-
i
^
+
2
j
^
-
k
^
a
1
→
.
b
1
→
×
b
2
→
=
0
i
^
+
2
j
^
-
3
k
^
.
-
i
^
+
2
j
^
-
k
^
=
0
+
4
+
3
=
7
a
2
→
.
b
1
→
×
b
2
→
=
2
i
^
+
6
j
^
+
3
k
^
.
-
i
^
+
2
j
^
-
k
^
=
-
2
+
12
-
3
=
7
Clearly,
a
1
→
.
b
1
→
×
b
2
→
=
a
2
→
.
b
1
→
×
b
2
→
Hence, the given lines are coplanar.
The equation of the plane containing the given lines is
r
→
.
b
1
→
×
b
2
→
=
a
1
→
.
b
1
→
×
b
2
→
⇒
r
→
.
-
i
^
+
2
j
^
-
k
^
=
0
i
^
+
2
j
^
-
3
k
^
.
-
i
^
+
2
j
^
-
k
^
⇒
r
→
.
-
i
^
+
2
j
^
-
k
^
=
7
⇒
r
→
.
i
^
-
2
j
^
+
k
^
+
7
=
0
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Similar questions
Q.
The vector equation of the plane containing the line
r
→
=
-
2
i
^
-
3
j
^
+
4
k
^
+
λ
3
i
^
-
2
j
^
-
k
^
and the point
i
^
+
2
j
^
+
3
k
^
is
(a)
r
→
·
i
^
+
3
k
^
=
10
(b)
r
→
·
i
^
-
3
k
^
=
10
(c)
r
→
·
3
i
^
+
k
^
=
10
(d) None of these
Q.
The angle between the lines
¯
¯
¯
r
=
(
2
¯
i
−
3
¯
j
+
¯
¯
¯
k
)
+
λ
(
¯
i
+
4
¯
j
+
3
¯
¯
¯
k
)
and
¯
¯
¯
r
=
(
¯
i
−
¯
j
+
2
¯
¯
¯
k
)
+
μ
(
¯
i
+
2
¯
j
−
3
¯
¯
¯
k
)
is
Q.
Show the each of the following triads of vectors are coplanar:
(i)
a
→
=
i
^
+
2
j
^
-
k
^
,
b
→
=
3
i
^
+
2
j
^
+
7
k
^
,
c
→
=
5
i
^
+
6
j
^
+
5
k
^
(ii)
a
→
=
-
4
i
^
-
6
j
^
-
2
k
^
,
b
→
=
-
i
^
+
4
j
^
+
3
k
^
,
c
→
=
-
8
i
^
-
j
^
+
3
k
^
(iii)
a
^
=
i
^
-
2
j
^
+
3
k
^
,
b
^
=
-
2
i
^
+
3
j
^
-
4
k
^
,
c
^
=
i
^
-
3
j
^
+
5
k
^
Q.
The distance between the line
¯
¯
¯
r
=
2
¯
i
−
2
¯
j
+
3
¯
¯
¯
k
+
λ
(
¯
i
−
¯
j
+
4
¯
¯
¯
k
)
and the plane
¯
¯
¯
r
.
(
¯
i
+
5
¯
j
+
¯
¯
¯
k
)
=
5
is
Q.
The distance between the line
¯
¯
¯
r
=
2
¯
i
−
2
¯
j
+
3
¯
¯
¯
k
+
λ
(
¯
i
−
¯
j
+
4
¯
¯
¯
k
)
and the plane
¯
¯
¯
r
(
¯
i
+
5
¯
j
+
¯
¯
¯
k
)
=
5
is
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