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Byju's Answer
Standard XII
Mathematics
Equation of Perpendicular from a Point on a Line
If n⃗ is a ...
Question
If
→
n
is a vector of magnitude
√
3
and is equally inclined with an acute angle with the coordinate axes. Find the vector and cartesian forms of equation of a plane which passes through
(
2
,
1
,
−
1
)
and is normal to
→
n
.
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Solution
Let
α
,
β
,
γ
angles of
¯
n
Given angels are equal So:
α
=
β
=
γ
ie:
cos
α
=
cos
β
=
cos
γ
So direction cosines
⇒
l
=
m
=
n
⇒
l
=
l
=
l
As we know
l
2
+
m
2
+
n
2
=
n
⇒
8
l
2
=
1
l
=
1
/
√
3
Given magnitude
=
√
3
¯
n
=
√
3
(
1
√
3
^
i
+
1
√
3
^
j
+
1
√
3
^
k
)
=
^
i
+
^
j
+
^
k
Given
(
2
,
1
,
−
1
)
⇒
¯
a
=
2
^
i
+
^
j
+
^
k
&
¯
n
=
^
i
+
^
j
+
^
k
get plane equation
¯
r
.
n
=
¯
a
.
¯
n
¯
r
(
^
i
+
^
j
+
^
k
)
=
(
2
^
i
+
^
j
−
^
k
)
(
^
i
+
^
j
+
^
k
)
=
2
+
1
−
1
¯
r
(
^
i
+
^
j
+
^
k
)
=
2
(
x
^
i
+
y
^
j
+
z
^
k
)
(
^
i
+
^
j
+
^
k
)
=
2
[
¯
r
=
x
^
i
+
y
^
j
+
z
^
k
]
x
+
y
+
z
=
2
As we want Question form
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