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

The vectors are mutually perpendicular

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The vector are linearly dependent

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The vectors are linearly independent

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the vectors are unit vectors

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Solution

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. $A = [\begin{matrix} X_{1} & X_{2} & X_{3} \end{matrix}] = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 2 & 3 & 1 5 & 6 & 4 \end{matrix} ⎤ ⎥ ⎦$
$| A | = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 2 & 3 & 1 5 & 6 & 4 \end{matrix} ∣ ∣ ∣ ∣$
$= 6 - 3 + (- 3) = 0$
$\Rightarrow$ linearly dependent.

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