Question

If $\overrightarrow{v_1}and\overrightarrow{v_2}$ 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

• A. True
• B. False

Answer 1

The correct option is B False

Let’s see the case where two vectors are collinear

$\overrightarrow{AB}and\overrightarrow{CD}$ are two collinear vectors. Now a linear combination of these vectors would be $R_1\overrightarrow{AB}+R_2\overrightarrow{CD}$ where $R_1$ and $R_2$ are scalars
$R_1\overrightarrow{AB}$ will be along $\overrightarrow{AB}$ or in opposite direction of $\overrightarrow{AB}$ depending upon $R_1$ is positive or negative.Same is the case with $R_2\overrightarrow{CD}$.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 $\overrightarrow{v_1}and\overrightarrow{v_2}$ to give any other vector in that plane, $\overrightarrow{v_1}and\overrightarrow{v_2}$ should not be collinear.
So the statement is false.