Prove that the vector $i - 3 j + 2 k, 2 i - 4 j - 4 k$ and $3 i + 2 j - k = 0$ are linearly independent.

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Solution

The vector $¯ ¯¯ ¯ A, ¯ ¯¯ ¯ B, ¯ ¯¯ ¯ C$ are linearly independent if $[¯ ¯¯ ¯ A, ¯ ¯¯ ¯ B, ¯ ¯¯ ¯ C] \neq 0$

here $(ˆ i - 3 ˆ j + 2 ˆ k) \cdot [(2 ˆ i - 4 ˆ j - 4 ˆ k) \times (3 ˆ i + 2 ˆ j - ˆ k)]$

$= (ˆ i - 3 ˆ j + 2 ˆ k) \cdot (12 ˆ i - 10 ˆ j + 16 ˆ k)$

$= 12 + 30 + 32 \neq 0$

$∴$ The given vectors are linearly independent

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