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Byju's Answer
Standard XII
Mathematics
Graph of Trigonometric Ratios
Find the valu...
Question
Find the value of θ ∈(0, π/2) for which vectors
a
→
=
sin
θ
i
^
+
cos
θ
j
^
and
b
→
=
i
^
-
3
j
^
+
2
k
^
are perpendicular.
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Solution
We have
a
→
=
sin
θ
i
^
+
cos
θ
j
^
and
b
→
=
i
^
-
3
j
^
+
2
k
^
It is given that the vectors are perpendicular.
⇒
a
→
.
b
→
=
0
⇒
sin
θ
-
3
cos
θ
=
0
⇒
sin
θ
=
3
cos
θ
⇒
tan
θ
=
3
⇒
θ
=
π
3
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0
Similar questions
Q.
If θ is an acute angle and the vector (sin θ)
i
^
+ (cos θ)
j
^
is perpendicular to the vector
i
^
-
3
j
^
,
then θ =
(a)
π
6
(b)
π
5
(c)
π
4
(d)
π
3
Q.
For what value of λ are the vectors
a
→
and
b
→
perpendicular to each other if
(i)
a
→
=
λ
i
^
+
2
j
^
+
k
^
and
b
→
=
4
i
^
-
9
j
^
+
2
k
^
(ii)
a
→
=
λ
i
^
+
2
j
^
+
k
^
and
b
→
=
5
i
^
-
9
j
^
+
2
k
^
(iii)
a
→
=
2
i
^
+
3
j
^
+
4
k
^
and
b
→
=
3
i
^
+
2
j
^
-
λ
k
^
(iv)
a
→
=
λ
i
^
+
3
j
^
+
2
k
^
and
b
→
=
i
^
-
j
^
+
3
k
^
Q.
(i) Find a vector of magnitude 49, which is perpendicular to both the vectors
2
i
^
+
3
j
^
+
6
k
^
and
3
i
^
-
6
j
^
+
2
k
^
.
(ii) Find a vector whose length is 3 and which is perpendicular to the vector
a
→
=
3
i
^
+
j
^
-
4
k
^
and
b
→
=
6
i
^
+
5
j
^
-
2
k
^
.
Q.
The value of
′
a
′
for which the points
A
,
B
,
C
with position vectors
2
¯
i
−
¯
j
+
¯
¯
¯
k
,
¯
i
−
3
¯
j
−
5
¯
¯
¯
k
and
a
¯
i
−
3
¯
j
+
¯
¯
¯
k
are the vertices of a right angled triangle with
∠
C
=
π
2
are :
Q.
Assertion :The system of linear equations
x
+
(
sin
θ
)
y
+
(
cos
θ
)
z
=
0
x
+
(
cos
θ
)
y
+
(
sin
θ
)
z
=
0
x
−
(
sin
θ
)
y
−
(
cos
θ
)
z
=
0
has a non-trivial solution for only one value of
θ
lying in the interval
(
0
,
π
2
)
. Reason: The equation in
θ
∣
∣ ∣
∣
cos
θ
sin
θ
cos
θ
sin
θ
cos
θ
sin
θ
cos
θ
−
sin
θ
−
cos
θ
∣
∣ ∣
∣
=
0
has only one solution lying in the interval
(
0
,
π
2
)
.
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