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Byju's Answer
Standard XII
Mathematics
Direction Cosines of a Line Passing through Two Points
Find the imag...
Question
Find the image of the point with position vector
3
i
^
+
j
^
+
2
k
^
in the plane
r
→
·
2
i
^
-
j
^
+
k
^
=
4
.
Also, find the position vectors of the foot of the perpendicular and the equation of the perpendicular line through
3
i
^
+
j
^
+
2
k
^
.
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Solution
Let
Q
be the image of the point
P
(3
i
^
+
j
^
+2
k
^
) in the plane
r
.
→
2
i
^
-
j
^
+
k
^
= 4
Since
PQ
passes through
P
and is normal to the given plane, it is parallel to the normal vector
2
i
^
-
j
^
+
k
^
.
So
,
the
equation of
PQ
is
r
→
=
3
i
^
+
j
^
+2
k
^
+
λ
2
i
^
-
j
^
+
k
^
As Q lies on PQ, let the position vector of Q be
3
+
2
λ
i
^
+
1
-
λ
j
^
+
2
+
λ
k
^
.
Let
R
be the mid-point of
PQ
. Then, the position vector of R is
3
+
2
λ
i
^
+
1
-
λ
j
^
+
2
+
λ
k
^
+
3
i
^
+
j
^
+2
k
^
2
=
6
+
2
λ
i
^
+
2
-
λ
j
^
+
4
+
λ
k
^
2
=
3
+
λ
i
^
+
1
-
λ
2
j
^
+
2
+
λ
2
k
^
Since
R
lies in the plane
r
.
→
2
i
^
-
j
^
+
k
^
= 4,
3
+
λ
i
^
+
1
-
λ
2
j
^
+
2
+
λ
2
k
^
.
2
i
^
-
j
^
+
k
^
= 4
⇒
6
+
2
λ
-
1
+
λ
2
+
2
+
λ
2
=
4
⇒
7
+
2
λ
+
λ
2
+
λ
2
=
4
⇒
14
+
6
λ
=
8
⇒
6
λ
=
8
-
14
⇒
λ
=
-
1
Putting
λ
=
-
1
in
Q
,
we
get
Q
=
3
+
2
(
-
1
)
i
^
+
1
-
(
-
1
)
j
^
+
2
+
(
-
1
)
k
^
=
i
^
+ 2
j
^
+
k
^
o
r
(
1
,
2
,
1
)
Therefore, by putting
λ
=
-
1
in
R
,
we get
R
=
3
+
(
-
1
)
i
^
+
1
-
(
-
1
)
2
j
^
+
2
+
(
-
1
)
2
k
^
=
2
i
^
+
3
2
j
^
+
3
2
k
^
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Similar questions
Q.
Find the vector equation of the plane passing through three points with position vectors
i
^
+
j
^
-
2
k
^
,
2
i
^
-
j
^
+
k
^
and
i
^
+
2
j
^
+
k
^
.
Also, find the coordinates of the point of intersection of this plane and the line
r
→
=
3
i
^
-
j
^
-
k
^
+
λ
2
i
^
-
2
j
^
+
k
^
.
Q.
Find the vector equation of a line passing through the point
A
whose position vector is
3
¯
i
+
¯
j
−
¯
¯
¯
k
and which is parallel to the vector
2
¯
i
−
¯
j
+
2
¯
¯
¯
k
.If
P
is a point of this line such that
A
P
=
15
,find the position vector of
P
.
Q.
Find a vector of magnitude
√
2
units and coplanar with vectors
3
i
−
j
−
k
and
i
+
j
−
2
k
and perpendicular to vector
2
i
+
2
j
+
k
.
Q.
Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector
2
i
^
+
3
j
^
+
4
k
^
to the plane
r
→
.
2
i
^
+
j
^
+
3
k
^
-
26
=
0
. Also find image of P in the plane.
Q.
If
i
−
j
+
2
k
,
2
i
+
j
−
k
and
3
i
−
j
+
2
k
are position vectors of vertices of a triangle ,then its area is
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