Video Transcript
True or False: The scalar triple product of vectors 𝐮, 𝐯, and 𝐰 results in a vector whose length is equal to the volume of the parallelepiped determined by vectors 𝐮, 𝐯, and 𝐰.
We begin by recalling the definition of the scalar triple product. This states that the scalar triple product of three vectors 𝐮, 𝐯, and 𝐰 is defined as shown. As its name suggests, it yields a scalar quantity. And this is equal to the determinant of the three-by-three matrix shown, where 𝑢 sub 𝑥, 𝑢 sub 𝑦, and 𝑢 sub 𝑧 are the components of vector 𝐮; 𝑣 sub 𝑥, 𝑣 sub 𝑦, and 𝑣 sub 𝑧 the components of vector 𝐯; and 𝑤 sub 𝑥, 𝑤 sub 𝑦, and 𝑤 sub 𝑧 the components of vector 𝐰.
Since this generates a scalar quantity and not a vector, we can already conclude that the statement is false. However, it is also worth noting the geometric meaning of the scalar triple product and how it links to this question. The volume of the parallelepiped determined by the vectors 𝐮, 𝐯, and 𝐰 is equal to the absolute value of the scalar triple product of vectors 𝐮, 𝐯, and 𝐰. This is the absolute value of the determinant of the three-by-three matrix shown. And as already mentioned, this generates a scalar quantity and not a vector.
