What is the volume of the parallelepiped defined by the vectors $$ \left[\begin{array}{c}-2 \ 4 \ 8\end{array} ight],\left[\begin{array}{c}-8 \ 5 \ 0\end{array} ight],\left[\begin{array}{l}0 \ 2 \ 9\end{array} ight] $$ Volume $$ = $$ Ex. 1

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What is the volume of the parallelepiped defined by the vectors $$ \left[\begin{array}{c}-2 \ 4 \ 8\end{array}ight],\left[\begin{array}{c}-8 \ 5 \ 0\end{array}ight],\left[\begin{array}{l}0 \ 2 \ 9\end{array}ight] $$ Volume $$ = $$ Ex. 1

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1. The volume $$ V $$ of the parallelepiped defined by vectors $$ \mathbf{a} $$, $$ \mathbf{b} $$, and $$ \mathbf{c} $$ is given by:
   $$
   V = \left| \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \right|
   $$

2. Given the vectors:
   $$
   \mathbf{a} = \begin{pmatrix} -2 \ 4 \ 8 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} -8 \ 5 \ 0 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 0 \ 2 \ 9 \end{pmatrix}
   $$

3. Compute the cross product $$ \mathbf{b} \times \mathbf{c} $$:
   $$
   \mathbf{b} \times \mathbf{c} = \begin{pmatrix}
   b_2 c_3 - b_3 c_2 \
   b_3 c_1 - b_1 c_3 \
   b_1 c_2 - b_2 c_1
   \end{pmatrix} = \begin{pmatrix}
   5 \cdot 9 - 0 \cdot 2 \
   0 \cdot 0 - (-8) \cdot 9 \
   -8 \cdot 2 - 5 \cdot 0
   \end{pmatrix} = \begin{pmatrix}
   45 \
   72 \
   -16
   \end{pmatrix}
   $$

4. Compute the dot product $$ \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) $$:
   $$
   \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = (-2)(45) + 4(72) + 8(-16)
   $$
   $$
   = -90 + 288 - 128 = 70
   $$

5. The volume of the parallelepiped is the absolute value of the triple scalar product:
   $$
   V = |70| = 70
   $$

Thus, the volume of the parallelepiped is $$ 70 $$.
The volume of the parallelepiped is 70.

What is the volume of the parallelepiped defined by the vectors
Volume Ex. 1

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What is the volume of the parallelepiped defined by the vectors Volume Ex. 1