Question

If, $2\overrightarrow{u}+3\overrightarrow{v}+4\overrightarrow{w}=5\overrightarrow{u}+6\overrightarrow{v}+7\overrightarrow{w}$, then the vectors $\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}$are linearly dependent.

Short answer

The above statement is true.

If,$2\overrightarrow{u}+3\overrightarrow{v}+4\overrightarrow{w}=5\overrightarrow{u}+6\overrightarrow{v}+7\overrightarrow{w}$ then the vectors $\overrightarrow{u.\overrightarrow{v}\overrightarrow{V}.}$are linearly dependent.

Step-by-step solution

Definition of linearly dependence vectors

Let V be a vector space over a field K. Let$v_1,v_2,v_3,...,v_n\in V$ , then if there exist scalars$a_1,a_2,a_3,...,a_n\in K$ , not all of them 0, such that

$a_1{v}_1+a_2{v}_{}+...a_n{v}_n=0$

Then the vectors $v_1,v_2,v_3,...,v_n$are called linearly dependent vectors.

Otherwise, linearly independent vectors.

To prove whether the vectors are linearly dependent or linearly independent.

We have given,

$2\overrightarrow{u}+3\overrightarrow{v}+4\overrightarrow{w}=5\overrightarrow{u}+6\overrightarrow{v}+7\overrightarrow{w}$

$\Rightarrow 3\overrightarrow{u}+3\overrightarrow{v}+3\overrightarrow{w}=0$

Here we can see that the scalars $a_1,a_2\text{and}{a}_3$are not equal to 0.

i.e. $a_1=3,a_2=3,a_3=3$

Thus, the vectors $\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}$are linearly dependent vectors.