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Question
If \[\vec{a}\], \[\vec{a}\], \[\vec{c}\] are non-coplanar vectors, prove that the following vectors are non-coplanar: \[\vec{a} + 2 \vec{b} + 3 \vec{c} , 2 \vec{a} + \vec{b} + 3 \vec{c}\text{ and }\vec{a} + \vec{b} + \vec{c}\]
Solution
Let if possible the following vector are coplanar. Then one of the vector is expressible in terms of the other two.
We have,
\[\vec{a} + 2 \vec{b} + 3 \vec{c} = x(2 \vec{a} + \vec{b} + 3 \vec{c} ) + y( \vec{a} + \vec{b} + \vec{c)} . \]
\[ = \vec{a} (2x + y) + \vec{b} (x + y) + \vec{c} (3x + y) . \]
\[ \Rightarrow 2x + y = 1, x + y = 2, 3x + y = 3 .\]
On solving the first two equations we get \[x = - 1, y = 3\].
Clearly the values of x, y does not satisfy the third equation.
Hence the given vectors are non-coplanar.
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