Video Transcript
True or False: The scalar triple product of vectors is equal to the determinant of the matrix formed from these vectors.
Let’s remind ourselves what the definition of the scalar triple product is. The scalar triple product of three vectors 𝐴, 𝐵, and 𝐶 is the dot product of the vector 𝐴 with the cross product of the vectors 𝐵 and 𝐶. In order to determine what this is actually equal to, let’s begin by writing down what the cross product of two three-dimensional vectors 𝐵 and 𝐶 is.
Suppose the vectors 𝐵 and 𝐶 have components 𝐵 sub 𝑥, 𝐵 sub 𝑦, 𝐵 sub 𝑧 and 𝐶 sub 𝑥, 𝐶 sub 𝑦, 𝐶 sub 𝑧, respectively. Then, their cross product is given by the vector whose 𝑖-component is 𝐵 sub 𝑦 𝐶 sub 𝑧 minus 𝐵 sub 𝑧 𝐶 sub 𝑦, whose 𝑗-component is negative 𝐵 sub 𝑥 𝐶 sub 𝑧 minus 𝐵 sub 𝑧 𝐶 sub 𝑥, and whose 𝑘-component is 𝐵 sub 𝑥 𝐶 sub 𝑦 minus 𝐵 sub 𝑦 𝐶 sub 𝑥.
Then, if we similarly define the vector 𝐴 to have components 𝐴 sub 𝑥, 𝐴 sub 𝑦, and 𝐴 sub 𝑧, respectively, then the dot product or scalar product of 𝐴 with the cross product of 𝐵 and 𝐶 is a scalar. It’s 𝐴 sub 𝑥 times 𝐵 sub 𝑦 𝐶 sub 𝑧 minus 𝐵 sub 𝑧 𝐶 sub 𝑦 minus 𝐴 sub 𝑦 times 𝐵 sub 𝑥 𝐶 sub 𝑧 minus 𝐵 sub 𝑧 𝐶 sub 𝑥 plus 𝐴 sub 𝑧 times 𝐵 sub 𝑥 𝐶 sub 𝑦 minus 𝐵 sub 𝑦 times 𝐶 sub 𝑥. But now, let’s think about what we know about finding the determinant of a three-by-three matrix.
We would find the determinant of a three-by-three matrix by multiplying each component in the top row by the determinant of the two-by-two matrix that remains when we eliminate that row and that column. So, for instance, we multiply 𝐴 sub 𝑥 by the determinant of the matrix 𝐵 sub 𝑦, 𝐵 sub 𝑧, 𝐶 sub 𝑦, 𝐶 sub 𝑧. And that gives us the first term in our scalar triple product.
In fact, if we continue, we do indeed find that our scalar triple product is equal to the determinant of this three-by-three matrix. And this three-by-three matrix is made up of rows which contain the elements of our vectors 𝐴, 𝐵, and 𝐶, respectively. And so, the answer here is true. The scalar triple product of vectors is equal to the determinant of the matrix formed from these vectors.
