overrightarrow{a}, overrightarrow{b}, overrig | vedclass

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(D) Since $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are non-coplanar,the vectors $\overrightarrow{P}, \overrightarrow{Q}, \overrigh

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infinite

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(D) Since $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are non-coplanar,the vectors $\overrightarrow{P}, \overrightarrow{Q}, \overrightarrow{R}$ are linearly dependent if and only if their scalar triple product is zero,i.e.,$[\overrightarrow{P}, \overrightarrow{Q}, \overrightarrow{R}] = 0$.This is equivalent to the determinant of the coefficients being zero:$\begin{vmatrix} 1 & 1 & 1 \ 4 & 3 & 4 \ 1 & \alpha & \beta \end{vmatrix} = 0$Expanding the determinant along the first row:$1(3\beta - 4\alpha) - 1(4\beta - 4) + 1(4\alpha - 3) = 0$$3\beta - 4\alpha - 4\beta + 4 + 4\alpha - 3 = 0$$-\beta + 1 = 0$$\beta = 1$Since the equation simplifies to $\beta = 1$ and is independent of $\alpha$,$\alpha$ can take any real value. Therefore,there are infinitely many possible values for $\alpha$.

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