Question

Show that $\vec a.(\vec b \times \vec c)$ is equal in magnitude to the volume of the parallelepiped formed on the three vectors $\vec a$,$\vec b$ and $\vec c$

Answer 1

Hint:The product of the area and altitude is known as the volume of the parallelepiped which is equal to the scalar triple product.Hence the vectors $\vec a$, $\vec b$ and $\vec c$ are parallelepiped. The vector product of the two vectors is known as the area of the parallelogram.

Complete step by step answer:
When the components are perpendicular to each other then it is called the rectangular components of the vector.

In the above diagram, the position vectors are,
$OA = \vec a$
$\Rightarrow OB = \vec b$
$\Rightarrow OC = \vec c$
We know that the parallelepiped volume is equal to the product of the parallelogram area and its height.
Hence we can write the volume of the parallelepiped $ = \vec a.(\vec b \times \vec c)$……………….. (1)
Let us consider $\vec n$ is the unit vector which is perpendicular to the $\vec b$ and $\vec c$. The $\vec n$ and $\vec c$ have the unique direction. So we can write,
$\vec b \times \vec c = bc\sin \theta \hat n$
Here the angle, $\theta = 90^\circ $. Finally,
$\vec b \times \vec c = bc\hat n$ …….(2)
Using the equation (1) the parallelepiped volume is $ = \vec a.(\vec b \times \vec c)$

Now we have to substitute the equation (2) into the equation (1) we get,
the parallelepiped volume is,
$\vec a.(bc\hat n) = abc\cos \theta \hat n$
The $\hat n$ is the unit vector so the angle becomes, $\theta = 0^\circ $. Hence the parallelepiped volume becomes, $abc$. And, the given parallelepiped volume is $abc $.From this explanation $\vec a.(\vec b \times \vec c)$ is equal to the parallelepiped volume.

Note:The three vectors are parallelepiped which means the vectors are present in the same plane which is a coplanar vector. In some particulars, the length of the vectors may be zero. The scalar is a component that has the only component which is the magnitude and the vector has the two components to specify which is a magnitude and the direction.