If vec{a} = hat{i} + hat{j} + hat{k},vec{b} = | vedclass

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Question

(C) The given determinant is the Gramian determinant of the vectors $\vec{a}, \vec{b}, \vec{c}$,which is equal to the square of the scalar triple prod

Vedclass · Accepted Answer

$16$

Vedclass · Answer

(C) The given determinant is the Gramian determinant of the vectors $\vec{a}, \vec{b}, \vec{c}$,which is equal to the square of the scalar triple product $[\vec{a} \vec{b} \vec{c}]^2$.First,we calculate the scalar triple product $[\vec{a} \vec{b} \vec{c}] = \vec{a} \cdot (\vec{b} 	imes \vec{c}) = \left| \begin{matrix} 1 & 1 & 1 \ 1 & -1 & 1 \ 1 & 2 & -1 \end{matrix} ight|$.Expanding the determinant along the first row:$[\vec{a} \vec{b} \vec{c}] = 1((-1)(-1) - (1)(2)) - 1((1)(-1) - (1)(1)) + 1((1)(2) - (-1)(1))$$= 1(1 - 2) - 1(-1 - 1) + 1(2 + 1)$$= 1(-1) - 1(-2) + 1(3)$$= -1 + 2 + 3 = 4$.Therefore,the value of the determinant is $[\vec{a} \vec{b} \vec{c}]^2 = 4^2 = 16$.

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