If the vectors \( (1.0, -1.0, 2.0), (7.0, 3.0, x) \) and \( (2.0, 3.0, 1.0) \) in \( \mathbb{R}^3 \) are linearly dependent, the value of \( x \) is _________.
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Correct Answer: 8
Solution and Explanation
For the vectors to be linearly dependent, the determinant of the matrix formed by these vectors must be zero. The matrix formed by these vectors is:
After simplifying and solving for \( x \), we get: \[ x = 8 \] Thus, the value of \( x \) is \( 8 \).
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