Question

Which one of the following describes the relationship among the three vectors,
$\hat{i}+\hat{j}+\hat{k},2\hat{i}+3\hat{j}+\hat{k},5\hat{i}+6\hat{j}+4\hat{k}$ ?

• A. The vectors are mutually perpendicular
• B. The vector are linearly dependent
• C. The vectors are linearly independent
• D. the vectors are unit vectors

Answer 1

The correct option is B The vector are linearly dependent

To check linear dependency, let us consider the given vectors as columns of matrix (A).
i.e. $A=\left[\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right]=\left[\begin{array}{ccc}1& 1& 1\\ 2& 3& 1\\ 5& 6& 4\end{array}\right]$
$|A|=\left|\begin{array}{ccc}1& 1& 1\\ 2& 3& 1\\ 5& 6& 4\end{array}\right|$
$=6-3+(-3)=0$
$\Rightarrow$ linearly dependent.