<p>If the volume of a parallelepiped whose adjacent edges are</p> <p><span class="math-tex">\(\rm \vec a\)</span> = 2î + 3ĵ + 4k̂</p> <p><span class="math-tex">\(\rm \vec b\)</span> = î + αĵ + 2k̂</p> <p><span class="math-tex">\(\rm \vec c\)</span> = î + 2ĵ + αk̂</p> <p>is 15 then α = ?</p>
Step-by-step Solution:
Volume: \[ \left| \begin{array}{ccc} 2 & 3 & 4 \\ 1 & \alpha & 2 \\ 1 & 2 & \alpha \end{array} \right| = 15 \] \[ \Rightarrow 2(\alpha^2 - 4) - 3(\alpha - 2) + 4(2 - \alpha) = 15 \] \[ \Rightarrow 2\alpha^2 - 7\alpha - 9 = 0 \] \[ \Rightarrow (\alpha + 1)(2\alpha - 9) = 0 \] Hence: \[ \alpha = -1, \quad \frac{9}{2} \]