Question

The vector is one of the vectors that are linearly dependent with the vector $2\hat{i}+3\hat{j}$

• A. $\hat{i}+\hat{j}$
• B. $4\hat{i}-\hat{j}$
• C. $4\hat{i}+6\hat{j}$

Answer 1

The correct option is C $4\hat{i}+6\hat{j}$

The given vector is $2\hat{i}+3\hat{j}$

Let it be $\overrightarrow{a}$

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are linearly dependent, then
$x\overrightarrow{a}+y\overrightarrow{b}=0$ where x and y are not simultaneously zero.

Now, among the given vectors, let's consider $4\hat{i}+6\hat{j}$

If it is linearly dependent with $2\hat{i}+3\hat{j}$, then

$x(2\hat{i}+3\hat{j})+y(4\hat{i}+6\hat{j})=0$ When x and y are not simultaneously 0

When x = 2 and y = -1, $x(2\hat{i}+3\hat{j})+y(4\hat{i}+6\hat{j})=0$

Thus, $4\hat{i}+6\hat{j}$ is linearly dependent with $2\hat{i}+3\hat{j}$