Question

If \[\vec{a} , \vec{b} , \vec{c}\] are three non-zero vectors, no two of which are collinear and the vector \[\vec{a} + \vec{b}\] is collinear with \[\vec{c} , \vec{b} + \vec{c}\] is collinear with \[\vec{a} ,\] then \[\vec{a} + \vec{b} + \vec{c} =\]

 

Options

• \[\vec{a}\]

   

• \[\vec{b}\]

   

• \[\vec{c}\]

   

• none of these

Answer 1

none of these
\[\vec{a} + \vec{b}\] is collinear with \[\vec{c}\]
\[\therefore \hspace{0.167em} \vec{a} + \vec{b} = x \vec{c} . . . . . (1)\]
where x is scalar and x ≠ 0.
\[\vec{b} + \vec{c}\] is collinear with \[\vec{a}\]
\[\vec{b} \hspace{0.167em} + \vec{c} = y \vec{a} . . . . . (2)\]
y is scalar and y ≠ 0
Substracting (2) from (1) we get,

\[\vec{a} - \vec{c} = x \vec{c} - y \vec{a}\]
\[\vec{a} (1 + y) = (1 + x) \vec{c}\]
As given

\[\vec{a} , \vec{c}\] are not collinear,
∴ 1 + y = 0 and 1 + x = 0
y = −1 and x = −1
Putting value of x in equation (1)
\[\begin{array}{l}\vec{a} + \vec{b} = - \vec{c} \\ \vec{a} + \vec{b} + \vec{c} = 0\end{array}\]