Show that $\vec{a} \cdot(\vec{b} \times \vec{c})$ is equal in magnitude to the volume of the parallelepiped formed by the three vectors $\vec{a}, \vec{b}$ and $\vec{c}$.
(N/A) The volume of a parallelepiped is defined as the product of the area of its base and its height.
Let the vectors representing the three adjacent edges of the parallelepiped be $\vec{a}, \vec{b}$,and $\vec{c}$.
The base of the parallelepiped is formed by vectors $\vec{b}$ and $\vec{c}$. The area of this base is given by the magnitude of the cross product: $Area = |\vec{b} \times \vec{c}|$.
The direction of the vector $\vec{b} \times \vec{c}$ is perpendicular to the base,i.e.,normal to the plane containing $\vec{b}$ and $\vec{c}$.
The height $h$ of the parallelepiped is the projection of vector $\vec{a}$ onto the direction of the normal vector $\vec{n} = \frac{\vec{b} \times \vec{c}}{|\vec{b} \times \vec{c}|}$.
Thus,$h = |\vec{a} \cdot \hat{n}| = \left| \vec{a} \cdot \frac{\vec{b} \times \vec{c}}{|\vec{b} \times \vec{c}|} \right|$.
The volume $V$ is given by $V = Area \times h = |\vec{b} \times \vec{c}| \times \left| \vec{a} \cdot \frac{\vec{b} \times \vec{c}}{|\vec{b} \times \vec{c}|} \right| = |\vec{a} \cdot (\vec{b} \times \vec{c})|$.
Therefore,the scalar triple product $\vec{a} \cdot (\vec{b} \times \vec{c})$ represents the volume of the parallelepiped.
Let the vectors representing the three adjacent edges of the parallelepiped be $\vec{a}, \vec{b}$,and $\vec{c}$.
The base of the parallelepiped is formed by vectors $\vec{b}$ and $\vec{c}$. The area of this base is given by the magnitude of the cross product: $Area = |\vec{b} \times \vec{c}|$.
The direction of the vector $\vec{b} \times \vec{c}$ is perpendicular to the base,i.e.,normal to the plane containing $\vec{b}$ and $\vec{c}$.
The height $h$ of the parallelepiped is the projection of vector $\vec{a}$ onto the direction of the normal vector $\vec{n} = \frac{\vec{b} \times \vec{c}}{|\vec{b} \times \vec{c}|}$.
Thus,$h = |\vec{a} \cdot \hat{n}| = \left| \vec{a} \cdot \frac{\vec{b} \times \vec{c}}{|\vec{b} \times \vec{c}|} \right|$.
The volume $V$ is given by $V = Area \times h = |\vec{b} \times \vec{c}| \times \left| \vec{a} \cdot \frac{\vec{b} \times \vec{c}}{|\vec{b} \times \vec{c}|} \right| = |\vec{a} \cdot (\vec{b} \times \vec{c})|$.
Therefore,the scalar triple product $\vec{a} \cdot (\vec{b} \times \vec{c})$ represents the volume of the parallelepiped.
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