Show that a(b × c) is equal in magnitude to the volume of the parallelepiped formed on the three vectors a, b and c.
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A parallelepiped with origin O and sides a, b, and c is shown in the following figure:
Volume of the given parallelepiped = abc $\vec{\text{O}}\text{C}=\vec{\text{a}}$ $\vec{\text{O}}\text{B}=\vec{\text{b}}$ $\vec{\text{O}}\text{C}=\vec{\text{c}}$ Let $\widehat{\text{n}}$ be a unit vector perpendicular to both b and c. Hence, $\widehat{\text{n}}$ and a have the same direction. $\therefore\ \vec{\text{b}}\times\vec{\text{c}}=\text{bc}\sin\theta\widehat{\text{n}}$ $=\text{bc}\sin90^{\circ}\widehat{\text{n}}$ $=\text{bc}\widehat{\text{n}}$ $\vec{\text{a}}\cdot\big(\vec{\text{b}}\times\vec{\text{c}}\big)$ $=\text{a}\cdot(\text{bc}\widehat{\text{n}})$ $=\text{abc}\cos\theta\widehat{\text{n}}$ $=\text{abc}\cos0^{\circ}$ $=\text{abc}$ = Volume of the parallelepiped.
Volume of the given parallelepiped = abc $\vec{\text{O}}\text{C}=\vec{\text{a}}$ $\vec{\text{O}}\text{B}=\vec{\text{b}}$ $\vec{\text{O}}\text{C}=\vec{\text{c}}$ Let $\widehat{\text{n}}$ be a unit vector perpendicular to both b and c. Hence, $\widehat{\text{n}}$ and a have the same direction. $\therefore\ \vec{\text{b}}\times\vec{\text{c}}=\text{bc}\sin\theta\widehat{\text{n}}$ $=\text{bc}\sin90^{\circ}\widehat{\text{n}}$ $=\text{bc}\widehat{\text{n}}$ $\vec{\text{a}}\cdot\big(\vec{\text{b}}\times\vec{\text{c}}\big)$ $=\text{a}\cdot(\text{bc}\widehat{\text{n}})$ $=\text{abc}\cos\theta\widehat{\text{n}}$ $=\text{abc}\cos0^{\circ}$ $=\text{abc}$ = Volume of the parallelepiped.
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