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Mathematics
Mathematics
Problem
Let
a
^
\hat{a}
a
^
and
b
^
\hat{b}
b
^
be two unit vectors such that
a
^
⋅
b
^
=
1
3
\hat{a} \cdot \hat{b} = \frac{1}{3}
a
^
⋅
b
^
=
3
1
and
a
^
×
b
^
=
c
^
\hat{a} \times \hat{b} = \hat{c}
a
^
×
b
^
=
c
^
. Also
F
⃗
=
α
a
^
+
β
b
^
+
γ
c
^
\vec{F} = \alpha \hat{a} + \beta \hat{b} + \gamma \hat{c}
F
=
α
a
^
+
β
b
^
+
γ
c
^
, where
α
,
β
,
γ
\alpha, \beta, \gamma
α
,
β
,
γ
are scalars. If
α
=
k
1
(
F
⃗
⋅
a
^
)
−
k
2
(
F
⃗
⋅
b
^
)
\alpha = k_1(\vec{F} \cdot \hat{a}) - k_2(\vec{F} \cdot \hat{b})
α
=
k
1
(
F
⋅
a
^
)
−
k
2
(
F
⋅
b
^
)
, then the value of
2
(
k
1
+
k
2
)
2(k_1 + k_2)
2
(
k
1
+
k
2
)
is a.
2
3
2\sqrt{3}
2
3
b.
3
\sqrt{3}
3
c. 3 d. 1
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