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Linear Combination of Vectors

If v⃗_⃗1⃗and ...

Question

If $_{1} a n d_{2}$ are two vectors in x – y plane. Then any vector in that plane can be obtained by the linear combination of these two vectors.The statement is

True

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False

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Solution

The correct option is B False
Let’s see the case where two vectors are collinear

$\to A B a n d \to C D$ are two collinear vectors. Now a linear combination of these vectors would be $R_{1} \to A B + R_{2} \to C D$ where $R_{1}$ and $R_{2}$ are scalars
$R_{1} \to A B$ will be along $\to A B$ or in opposite direction of $\to A B$ depending upon $R_{1}$ is positive or negative.Same is the case with $R_{2} \to C D$ .So their addition will be either along the same line or in the exactly opposite direction of that line. Any linear combination would not give a vector which does not lie on the mentioned line. So for the linear combination of $_{1} a n d_{2}$ to give any other vector in that plane, $_{1} a n d_{2}$ should not be collinear.
So the statement is false.

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