Video Transcript
Find the volume of the
parallelepiped with the adjacent sides 𝐮 is equal to one, one, three; 𝐯 is the
vector two, one, four; and 𝐰 is the vector five, one, negative two.
The parallelepiped as defined is
spun by the vectors 𝐮, 𝐯, and 𝐰. And we know that to find the volume
of such a parallelepiped, we can use the scalar triple product. That is, the volume of the
parallelepiped with adjacent sides 𝐮, 𝐯, and 𝐰 is the magnitude of the scalar
triple product. We also know that the scalar triple
product is the determinant of the matrix whose rows are the elements of the vectors
𝐮, 𝐯, and 𝐰. So in fact, the volume is the
magnitude of this.
In our case, then this is the
magnitude of the determinant of the matrix whose elements are one, one, three; two,
one, four; and five, one, negative two. That is where the rows are our
vectors 𝐮, 𝐯, and 𝐰. That is one times the determinant
of the two-by-two matrix with elements one, four, one, negative two minus one times
the two-by-two matrix with elements two, four, five, and negative two plus three
times the determinant of the two-by-two matrix with elements two, one, five, and
one.
And using the fact that the
determinant of a two-by-two matrix with elements 𝑎, 𝑏, 𝑐, 𝑑 is 𝑎𝑑 minus 𝑏𝑐,
this evaluates to the magnitude of negative six plus 24 minus nine, which is
nine. The volume of the parallelepiped
with adjacent sides 𝐮, 𝐯, and 𝐰 is therefore nine cubic units.
