Question

Which one of the following describes the relationship among the three vectors, $\hat{i}+\hat{j}+\hat{k},2\hat{i}+3\hat{j}+\hat{k},and5\hat{i}+6\hat{j}+4\hat{k}$?

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

Answer 1

The correct option is B The vectors are linearly dependent

Let $X_1=(\overline{i}+\overline{j}+\overline{k)}={\left[\begin{array}{ccc}1& 1& 1\end{array}\right]}^{\prime }$
$X_2=(2\overline{i}+3\overline{j}+\overline{k})={\left[\begin{array}{ccc}2& 3& 1\end{array}\right]}^{\prime }$
$X_3=(5\overline{i}+6\overline{j}+4\overline{k})={\left[\begin{array}{ccc}5& 6& 4\end{array}\right]}^{\prime }$
To check linear dependency, let us consider the given vectors as columns of matrix (A).
i.e. $A=\left[\begin{array}{ccc}{X}_1& X_2& X_3\end{array}\right]=\left[\begin{array}{ccc}1& 2& 5\\ 1& 3& 6\\ 1& 1& 4\end{array}\right]$
$\left|A\right|=0$
So given vectors are linearly dependent.