Video Transcript
The parallelepiped on vectors
negative two, negative two, 𝑚; two, zero, negative two; and negative five, one,
zero has volume 48. What can 𝑚 be?
We begin by recalling that a
parallelepiped is a three-dimensional shape of which each face is a
parallelogram. And its volume is calculated by
finding the absolute value of the scalar triple product of the three vectors 𝐀, 𝐁,
and 𝐂 as shown. In this question, we will let
vector 𝐀 equal negative two, negative two, 𝑚. Vector 𝐁 is equal to two, zero,
negative two. And vector 𝐂 is equal to negative
five, one, zero. It is important to note, however,
that the order we take these vectors doesn’t matter.
We recall that the scalar triple
product of three vectors is equal to the determinant of a three-by-three matrix. We will populate this matrix by
adding the components of vector 𝐀 to the top row, the components of vector 𝐁 to
the middle row, and the components of vector 𝐂 to the bottom row. The scalar triple product of
vectors 𝐀, 𝐁, and 𝐂 is the determinant of the three-by-three matrix negative two,
negative two, 𝑚, two, zero, negative two, negative five, one, zero.
Expanding over the top row, this is
equal to negative two multiplied by the determinant of the two-by-two matrix zero,
negative two, one, zero minus negative two multiplied by the determinant of the
two-by-two matrix two, negative two, negative five, zero plus 𝑚 multiplied by the
determinant of the two-by-two matrix two, zero, negative five, one. The determinant of a two-by-two
matrix with elements 𝑎, 𝑏, 𝑐, 𝑑 is equal to 𝑎𝑑 minus 𝑏𝑐. So our expression simplifies to
negative two multiplied by two plus two multiplied by negative 10 plus 𝑚 multiplied
by two. This is equal to negative four
minus 20 plus two 𝑚, which simplifies to two 𝑚 minus 24. The scalar triple product of
vectors 𝐀, 𝐁, and 𝐂 is two 𝑚 minus 24.
Since the volume of a
parallelepiped is equal to the absolute value of the scalar triple product and we
are told that the volume is equal to 48, we have the absolute value of two 𝑚 minus
24 equals 48. This gives us two possible
solutions: either two 𝑚 minus 24 equals 48, or two 𝑚 minus 24 equals negative
48. Adding 24 to both sides of both
equations, we have two 𝑚 equals 72 and two 𝑚 equals negative 24. We can then divide through by two
giving us 𝑚 is equal to 36 or 𝑚 is equal to negative 12. The two values that 𝑚 can take
such that the volume of the given parallelepiped is 48 are 36 and negative 12.
