The following figure shows a parallelepiped with sides a, b, c and origin at O.
![](https://search-static.byjusweb.com/question-images/scholr/scholr-acadecraft/Images/HTML_46124_1.JPG)
The expression for volume V of a parallelepiped is given as,
V=abc(1)
Let n ^ be the unit vector perpendicular to vectors b and c, therefore n ^ and a will have the same direction.
The cross product of vector b and c is given as,
b×c=( bcsinθ ) n ^
Here, θ is the angle between b and c.
Substitute the values in the above expression.
b×c=cbsin( 90 ∘ ) n ^ =bc n ^
The dot product of a with b×c is given as,
a⋅( b×c )
Substitute the values in the above expression.
a⋅( b×c )=a⋅( ( bc ) n ^ ) =abccos θ ′
Here, the value of θ ′ will be 0°as the direction of n ^ and a is same.
Substitute the values in the above expression.
a⋅( b×c )=abccos0° =abc [ Since cos0°=1 ] (2)
Hence, it is proved that volume formed a, b and c is equal to the magnitude of a⋅( b×c ).