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Tardigrade
Question
Mathematics
If the vector a =3 hat j +4 hat k is the sum of two vectors a 1 and a 2, vector a 1 is parallel to b = hat i + hat j and vector a 2 is perpendicular to b, then a 1=
Q. If the vector
a
=
3
j
^
+
4
k
^
is the sum of two vectors
a
1
and
a
2
, vector
a
1
is parallel to
b
=
i
^
+
j
^
and vector
a
2
is perpendicular to
b
, then
a
1
=
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A
2
1
(
i
^
+
j
^
)
B
3
1
(
i
^
+
j
^
)
C
3
2
(
i
^
+
j
^
)
D
2
3
(
i
^
+
j
^
)
Solution:
Given,
a
=
3
j
^
+
4
k
^
and
b
=
i
^
+
j
^
a
=
a
1
+
a
2
a
1
is parallel to
b
.
∴
a
1
=
λb
=
λ
(
i
^
+
j
^
)
a
2
is perpendicular to
b
.
∴
a
2
⋅
b
=
0
(
a
−
a
1
)
⋅
b
=
0
(
3
j
^
+
4
k
^
−
λ
(
i
^
+
j
^
))
⋅
(
i
^
+
j
^
)
=
0
(
−
λ
i
^
+
(
3
−
λ
)
j
^
+
4
k
^
)
⋅
(
i
^
+
j
^
)
=
0
⇒
−
λ
+
3
−
λ
=
0
⇒
λ
=
2
3
∴
a
1
=
2
3
(
i
^
+
j
^
)