Question :-
[tex]\rm \: The \: vectors \: \vec{a} - \vec{b}, \: \vec{b} - \vec{c}, \: \vec{c} - \vec{a} \: are - - - - \\ [/tex]
[tex]\large\underline{\sf{Solution-}}[/tex]
Consider
[tex]\rm \: [\vec{a} - \vec{b} \: \: \vec{b} - \vec{c} \: \: \vec{c} - \vec{a}] \\ [/tex]
can be rewritten as
[tex]\rm \: = (\vec{a} - \vec{b}).\bigg((\vec{b} - \vec{c}) \times (\vec{c} - \vec{a}) \bigg) \\ [/tex]
[tex]\rm \: = (\vec{a} - \vec{b}).\bigg(\vec{b} \times \vec{c} - \vec{b} \times \vec{a} - \vec{c} \times \vec{c} + \vec{c} \times \vec{a} \bigg) \\ [/tex]
We know,
[tex]\boxed{ \rm{ \:\vec{a} \times \vec{a} = 0 \: }} \\ [/tex]
So, using this, we get
[tex]\rm \: = (\vec{a} - \vec{b}).\bigg(\vec{b} \times \vec{c} - \vec{b} \times \vec{a} + \vec{c} \times \vec{a} \bigg) \\ [/tex]
[tex]\rm \: =\vec{a}.(\vec{b} \times \vec{c}) - \vec{a}.(\vec{b} \times \vec{a}) + \vec{a}.(\vec{c} \times \vec{a}) - \vec{b}.(\vec{b} \times \vec{c}) + \vec{b}.(\vec{b} \times \vec{a}) - \vec{b}.(\vec{c} \times \vec{a}) \\ [/tex]
[tex]\rm \: =\vec{a}.(\vec{b} \times \vec{c}) - 0 + 0 - 0 + 0 - \vec{b}.(\vec{c} \times \vec{a}) \\ [/tex]
[tex]\rm \: =\vec{a}.(\vec{b} \times \vec{c}) - \vec{a}.(\vec{b} \times \vec{c}) \\ [/tex]
[tex]\rm \: = \: 0[/tex]
Thus,
[tex]\rm\implies \:\rm \: [\vec{a} - \vec{b} \: \: \vec{b} - \vec{c} \: \: \vec{c} - \vec{a}] = 0 \\ [/tex]
[tex]\rm\implies \: \vec{a} - \vec{b}, \: \vec{b} - \vec{c}, \: \vec{c} - \vec{a} \: are \: coplanar \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Formulae Used :-
[tex]\boxed{ \rm{ \:[\vec{a} \: \: \vec{b} \: \: \vec{c}] \: = \: \vec{a}.(\vec{b} \times \vec{c}) \: \: }} \\ [/tex]
[tex]\boxed{ \rm{ \: \:\vec{a}.(\vec{b} \times \vec{c}) = \vec{b}.(\vec{c} \times \vec{a}) = \vec{c}.(\vec{a} \times \vec{b}) }} \\ [/tex]
[tex]\boxed{ \rm{ \: \:[\vec{a} \: \: \vec{b} \: \: \vec{c}] = 0 \: \rm\implies \:\vec{a},\vec{b},\vec{c} \: are \: coplanar \: }} \\ [/tex]
[tex]\boxed{ \rm{ \: \:[\vec{a} \: \: \vec{a} \: \: \vec{b}] = 0 \: \: }} \\ [/tex]
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Answers & Comments
Question :-
[tex]\rm \: The \: vectors \: \vec{a} - \vec{b}, \: \vec{b} - \vec{c}, \: \vec{c} - \vec{a} \: are - - - - \\ [/tex]
[tex]\large\underline{\sf{Solution-}}[/tex]
Consider
[tex]\rm \: [\vec{a} - \vec{b} \: \: \vec{b} - \vec{c} \: \: \vec{c} - \vec{a}] \\ [/tex]
can be rewritten as
[tex]\rm \: = (\vec{a} - \vec{b}).\bigg((\vec{b} - \vec{c}) \times (\vec{c} - \vec{a}) \bigg) \\ [/tex]
[tex]\rm \: = (\vec{a} - \vec{b}).\bigg(\vec{b} \times \vec{c} - \vec{b} \times \vec{a} - \vec{c} \times \vec{c} + \vec{c} \times \vec{a} \bigg) \\ [/tex]
We know,
[tex]\boxed{ \rm{ \:\vec{a} \times \vec{a} = 0 \: }} \\ [/tex]
So, using this, we get
[tex]\rm \: = (\vec{a} - \vec{b}).\bigg(\vec{b} \times \vec{c} - \vec{b} \times \vec{a} + \vec{c} \times \vec{a} \bigg) \\ [/tex]
[tex]\rm \: =\vec{a}.(\vec{b} \times \vec{c}) - \vec{a}.(\vec{b} \times \vec{a}) + \vec{a}.(\vec{c} \times \vec{a}) - \vec{b}.(\vec{b} \times \vec{c}) + \vec{b}.(\vec{b} \times \vec{a}) - \vec{b}.(\vec{c} \times \vec{a}) \\ [/tex]
[tex]\rm \: =\vec{a}.(\vec{b} \times \vec{c}) - 0 + 0 - 0 + 0 - \vec{b}.(\vec{c} \times \vec{a}) \\ [/tex]
[tex]\rm \: =\vec{a}.(\vec{b} \times \vec{c}) - \vec{a}.(\vec{b} \times \vec{c}) \\ [/tex]
[tex]\rm \: = \: 0[/tex]
Thus,
[tex]\rm\implies \:\rm \: [\vec{a} - \vec{b} \: \: \vec{b} - \vec{c} \: \: \vec{c} - \vec{a}] = 0 \\ [/tex]
[tex]\rm\implies \: \vec{a} - \vec{b}, \: \vec{b} - \vec{c}, \: \vec{c} - \vec{a} \: are \: coplanar \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Formulae Used :-
[tex]\boxed{ \rm{ \:[\vec{a} \: \: \vec{b} \: \: \vec{c}] \: = \: \vec{a}.(\vec{b} \times \vec{c}) \: \: }} \\ [/tex]
[tex]\boxed{ \rm{ \: \:\vec{a}.(\vec{b} \times \vec{c}) = \vec{b}.(\vec{c} \times \vec{a}) = \vec{c}.(\vec{a} \times \vec{b}) }} \\ [/tex]
[tex]\boxed{ \rm{ \: \:[\vec{a} \: \: \vec{b} \: \: \vec{c}] = 0 \: \rm\implies \:\vec{a},\vec{b},\vec{c} \: are \: coplanar \: }} \\ [/tex]
[tex]\boxed{ \rm{ \: \:[\vec{a} \: \: \vec{a} \: \: \vec{b}] = 0 \: \: }} \\ [/tex]