Question #1bfc9

2 Answers
Jan 23, 2018

60 units cubed.


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

#|(3,0,0),(0,4,0),(0,0,5)| = 3(4*5-0)-0(0-0)+0(0-0) = 3(20)=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.


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 .

enter image source here

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 #(vec b X vec c)# 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 #(vec a . (vec b X vec c))# .

#vec u = <3,0,0> , vec v = <0,4,0> , vec w = <0,0,5>#.

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

The final answer is #(3,0,0). (20,0,0) = 60# cubic units.

Hope this helps!