Show that $a \cdot( b \times c )$ is equal in magnitude to the volume of the parallelepiped formed on the three vectors, $a, b$ and $c$.
Show that $a \cdot( b \times c )$ is equal in magnitude to the volume of the parallelepiped formed on the three vectors, $a, b$ and $c$.
Volume of the given parallelepiped $=a b c$
$\overrightarrow{ OC }=\vec{a}$
$\overrightarrow{ OB }=\vec{b}$
$\overrightarrow{ OC }=\vec{c}$
Let $\hat{ n }$ be a unit vector perpendicular to both $b$ and $c .$ Hence, $\quad \hat{ n }$ and $a$ have the same direction. $\therefore \vec{b} \times \vec{c}=b c \sin \theta \hat{ n }$
$=b c \sin 90^{\circ} \hat{ n }$
$=b c \hat{n}$
$\vec{a} \cdot(\vec{b} \times \vec{c})$
$=a \cdot(b c \hat{ n })$
$=a b c \cos \theta \hat{ n }$
$=a b c \cos 0^{\circ}$
$=a b c$
$=$ Volume of the parallelepiped
Similar Questions
If $\left| {\vec A } \right|\, = \,2$ and $\left| {\vec B } \right|\, = \,4$ then match the relation in Column $-I$ with the angle $\theta $ between $\vec A$ and $\vec B$ in Column $-II$.
Column $-I$
Column $-II$
$(a)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,0$
$(i)$ $\theta = \,{30^o}$
$(b)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,8$
$(ii)$ $\theta = \,{45^o}$
$(c)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,4$
$(iii)$ $\theta = \,{90^o}$
$(d)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,4\sqrt 2$
$(iv)$ $\theta = \,{0^o}$
If $\left| {\vec A } \right|\, = \,2$ and $\left| {\vec B } \right|\, = \,4$ then match the relation in Column $-I$ with the angle $\theta $ between $\vec A$ and $\vec B$ in Column $-II$.
| Column $-I$ | Column $-II$ |
| $(a)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,0$ | $(i)$ $\theta = \,{30^o}$ |
| $(b)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,8$ | $(ii)$ $\theta = \,{45^o}$ |
| $(c)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,4$ | $(iii)$ $\theta = \,{90^o}$ |
| $(d)$ $\left| {\vec A \, \times \,\,\vec B } \right|\, = \,\,4\sqrt 2$ | $(iv)$ $\theta = \,{0^o}$ |
Explain the kinds of multiplication operations for vectors.
Explain the kinds of multiplication operations for vectors.
Dot product of two mutual perpendicular vector is
Dot product of two mutual perpendicular vector is
The modulus of the vector product of two vectors is $\frac{1}{\sqrt{3}}$ times their scalar product. The angle between vectors is
The area of the triangle formed by $2\hat i + \hat j - \hat k$ and $\hat i + \hat j + \hat k$ is
The area of the triangle formed by $2\hat i + \hat j - \hat k$ and $\hat i + \hat j + \hat k$ is