Video Transcript
If 𝐯, 𝐰, and 𝐮 are three linearly independent vectors in the three-dimensional real plane, then the volume of the parallelepiped determined by vectors 𝐯, 𝐰, and 𝐮 is what.
In order to answer this question, we recall the geometric meaning of the scalar triple product. The absolute value of the scalar triple product of three vectors is the volume of the parallelepiped spanned by the three vectors. If the three vectors are 𝐀, 𝐁, and 𝐂, then the volume is equal to the absolute value of the scalar triple product of 𝐀, 𝐁, and 𝐂. In this question, our three vectors are 𝐯, 𝐰, and 𝐮, which means that the volume of the parallelepiped is equal to the absolute value of the scalar triple product of vectors 𝐯, 𝐰, and 𝐮.
It is worth noting that if the vectors 𝐯, 𝐰, and 𝐮 had components as shown, then the scalar triple product of vectors 𝐯, 𝐰, and 𝐮 is equal to the determinant of the three-by-three matrix 𝑣 sub 𝑥, 𝑣 sub 𝑦, 𝑣 sub 𝑧, 𝑤 sub 𝑥, 𝑤 sub 𝑦, 𝑤 sub 𝑧, 𝑢 sub 𝑥, 𝑢 sub 𝑦, 𝑢 sub 𝑧. And the volume of the parallelepiped will be equal to the absolute value of this determinant.
