Answer this ques with solution.
Answers
Given :
To find :
Angle between vector a and b.
Solution :
Now, comparing both sides :-
Answer :-
option (c)5π/6 is correct
Step-by-step explanation:
Given :
\begin{gathered} \rm| \vec{a} | = 1 \\ \\ \rm| \vec{b} | = 1 \\ \\ \rm| \vec{c} | = 1\end{gathered}
∣
a
∣=1
∣
b
∣=1
∣
c
∣=1
\rm \vec{a} \times (\vec{b} \times\vec{c} ) = \dfrac{ \sqrt{3} }{2} (\vec{b} + \vec{c} )
a
×(
b
×
c
)=
2
3
(
b
+
c
)
To find :
Angle between vector a and b.
Solution :
\begin{gathered}\rm \implies\vec{a} \times (\vec{b} \times\vec{c} ) = \dfrac{ \sqrt{3} }{2} (\vec{b} + \vec{c} ) \\ \\ \rm \implies(\vec{a} .\vec{c} ) \vec{b} - (\vec{a} . \vec{b}) \vec{c} = \dfrac{ \sqrt{3} }{2} (\vec{b} + \vec{c} )\end{gathered}
⟹
a
×(
b
×
c
)=
2
3
(
b
+
c
)
⟹(
a
.
c
)
b
−(
a
.
b
)
c
=
2
3
(
b
+
c
)
Now, comparing both sides :-
\begin{gathered}\rm \implies - (\vec{a} . \vec{b}) \vec{c} = \bigg( \dfrac{ \sqrt{3} }{2} \bigg)\vec{c} \\ \\ \rm \implies - (\vec{a} . \vec{b}) = \dfrac{ \sqrt{3} }{2} \\ \\ \rm \implies \vec{a} . \vec{b} = - \dfrac{ \sqrt{3} }{2} \\ \\ \rm \implies| \vec{a} | |\vec{b} |cos \theta = - \dfrac{ \sqrt{3} }{2} \\ \\ \rm \implies1 \times 1 \times cos \theta = - \dfrac{ \sqrt{3} }{2} \\ \\ \rm \implies cos \theta = - \dfrac{ \sqrt{3} }{2} \\ \\ \rm \implies \theta = \dfrac{ {5\pi} }{6} \end{gathered}
⟹−(
a
.
b
)
c
=(
2
3
)
c
⟹−(
a
.
b
)=
2
3
⟹
a
.
b
=−
2
3
⟹∣
a
∣∣
b
∣cosθ=−
2
3
⟹1×1×cosθ=−
2
3
⟹cosθ=−
2
3
⟹θ=
6
5π
Answer :-
option (c)5π/6 is correct