If the volume of a parallelopiped whose adjacent edges are \( \mathbf{a}=2 \hat{\mathbf{i}}+3 \h...

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If the volume of a parallelopiped whose adjacent edges are \( \mathbf{a}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+\alpha \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \mathbf{c}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\alpha \hat{\mathbf{k}} \) is 15, then find the value of \( \alpha \), where \( \alpha0 \).
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