# Question #1bfc9

Jan 23, 2018

60 units cubed.

#### Explanation:

The volume of the parallelepiped is the absolute value of the scalar triple product, so in this case it's the absolute value of

$| \left(3 , 0 , 0\right) , \left(0 , 4 , 0\right) , \left(0 , 0 , 5\right) | = 3 \left(4 \cdot 5 - 0\right) - 0 \left(0 - 0\right) + 0 \left(0 - 0\right) = 3 \left(20\right) = 60$,

which is positive, so the volume is 60 units cubed.

In this case the vectors are all orthogonal to each other so this is a rectangular prism with sides 3, 4, and 5, so the volume is lengthwidthheight, which again gives 60 units cubed.

Jan 23, 2018

parallelepiped is what you get when you make a cuboid with parallelograms instead of rectangles.

#### Explanation:

This might go long !

Speaking geometrically , the volume of the parallelepiped is similar to that of a cuboid (cube is also a cuboid made of perfect rectangles , i.e , squares). Its the product of length , width and height .

See the image above . If it were a cuboid , the volume would be
bca . But this is not the case in parallel-boid (that's made up for sure ! ) . Do you notice how i said volume = length * width * height?
What height means is the perpendicular distance between the base and opposite face , that is marked 'h' in the picture above.
Also note that $\left(\vec{b} X \vec{c}\right)$ gives a vector in the direction perpendicular to base .

Now using the three vectors given , you can compute the box product (also known as triple product).
The notation is [a,b,c] . The operation is $\left(\vec{a} . \left(\vec{b} X \vec{c}\right)\right)$ .

$\vec{u} = < 3 , 0 , 0 > , \vec{v} = < 0 , 4 , 0 > , \vec{w} = < 0 , 0 , 5 >$.

volume = $\left[\vec{u} , \vec{v} , \vec{w}\right]$ = $\left[\vec{u} . \left(\vec{v} X \vec{w}\right)\right]$
I am skipping the calculation hoping you know how to calculate dot products and cross products .

The final answer is $\left(3 , 0 , 0\right) . \left(20 , 0 , 0\right) = 60$ cubic units.

Hope this helps!