Question
Calculate the volume of a parallelepiped whose sides are given by the vectors
a = 3i +2j +k,= – i +3j and c = 2i+2j + 5k.
Answer :
Word Count : 281
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The volume of a parallelepiped formed by three vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) is given by the absolute value of the scalar triple product: \[ V = \left| \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \right| \] Given vectors: \[ \mathbf{a} = 3\hat{i} + 2\hat{j} + \hat{k} \] \[ \mathbf{b} = -\hat{i} + 3\hat{j} + 0\hat{k} \] \[ \mathbf{c} = 2\hat{i} + 2\hat{j} + 5\hat{k} \] ### Step 1: Compute \(\mathbf{b} \times \mathbf{c}\) The cross product is computed using the determinant of the matrix: \[ \mathbf{b} \times \mathbf{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -1 & 3 & 0 \\ 2
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