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Byju's Answer
Standard XII
Mathematics
Coordinate Axes in Three Dimensional Space
The vector eq...
Question
The vector equation of the line of intersection of the planes
r
.
(
i
+
2
j
+
3
k
)
=
0
and
r
.
(
3
i
+
2
j
+
k
)
=
0
is
A
r
=
λ
(
i
+
2
j
+
k
)
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B
r
=
λ
(
i
−
2
j
+
k
)
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C
r
=
λ
(
i
+
2
j
−
3
k
)
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D
None of these
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Solution
The correct option is
A
r
=
λ
(
i
−
2
j
+
k
)
The line of intersection of the planes
r
.
(
i
+
2
j
+
3
k
)
=
0
and
r
.
(
3
i
+
2
j
+
k
)
=
0
is parallel to the vector
(
i
+
2
j
+
3
k
)
×
(
3
i
+
2
j
+
k
)
=
−
4
i
+
8
j
−
4
k
Since both the planes pass through the origin, therefore their line of intersection will also pass through the origin.
Thus, the required line passes through the origin and is parallel to the vector
−
4
i
+
8
j
−
4
k
Hence, its equation is
r
=
0
+
λ
′
(
−
4
i
+
8
j
−
4
k
)
⇒
r
=
λ
(
i
−
2
j
+
k
)
where
λ
=
−
4
λ
′
Suggest Corrections
0
Similar questions
Q.
The equation of the plane
r
→
=
i
^
-
j
^
+
λ
i
^
+
j
^
+
k
^
+
μ
i
^
-
2
j
^
+
3
k
^
in scalar product form is
(a)
r
→
·
5
i
^
-
2
j
^
-
3
k
^
=
7
(b)
r
→
·
5
i
^
+
2
j
^
-
3
k
^
=
7
(c)
r
→
·
5
i
^
-
2
j
^
+
3
k
^
=
7
(d) None of these
Q.
Find the vector equations of the following planes in scalar product form
r
→
·
n
→
=
d
:
(i)
r
→
=
2
i
^
-
k
^
+
λ
i
^
+
μ
i
^
-
2
j
^
-
k
^
(ii)
r
→
=
1
+
s
-
t
t
^
+
2
-
s
j
^
+
3
-
2
s
+
2
t
k
^
(iii)
r
→
=
i
^
+
j
^
+
λ
i
^
+
2
j
^
-
k
^
+
μ
-
i
^
+
j
^
-
2
k
^
(iv)
r
→
=
i
^
-
j
^
+
λ
i
^
+
j
^
+
k
^
+
μ
4
i
^
-
2
j
^
+
3
k
^
Q.
Find the vector equation of the following planes in non-parametric form.
(i)
r
→
=
λ
-
2
μ
i
^
+
3
-
μ
j
^
+
2
λ
+
μ
k
^
(ii)
r
→
=
2
i
^
+
2
j
^
-
k
^
+
λ
i
^
+
2
j
^
+
3
k
^
+
μ
5
i
^
-
2
j
^
+
7
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.
Find the shortest distance between the lines
(i)
r
→
=
i
^
+
2
j
^
+
k
^
+
λ
i
^
-
j
^
+
k
^
and
,
r
→
=
2
i
^
-
j
^
-
k
^
+
μ
2
i
^
+
j
^
+
2
k
^
(ii)
x
+
1
7
=
y
+
1
-
6
=
z
+
1
1
and
x
-
3
1
=
y
-
5
-
2
=
z
-
7
1
(iii)
r
→
=
i
^
+
2
j
^
+
3
k
^
+
λ
i
^
-
3
j
^
+
2
k
^
and
r
→
=
4
i
^
+
5
j
^
+
6
k
^
+
μ
2
i
^
+
3
j
^
+
k
^
(iv)
r
→
=
6
i
^
+
2
j
^
+
2
k
^
+
λ
i
^
-
2
j
^
+
2
k
^
and
r
→
=
-
4
i
^
-
k
^
+
μ
3
i
^
-
2
j
^
-
2
k
^
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