Question Video: Calculating the Scalar Triple Product Mathematics

Video Transcript

Given that the vector 𝐀 is equal to one, two, four; 𝐁 is equal to two, four, negative one; and 𝐂 is equal to negative one, four, two, find the scalar triple product of 𝐀, 𝐁, and 𝐂 plus the scalar triple product of 𝐁, 𝐂, and 𝐀 plus the scalar triple product of 𝐂, 𝐀, and 𝐁.

When we read each part of this sum, we described it as the scalar triple product. Now it is exactly as it reads. It’s the dot product of the vector 𝐀 here with the cross product of the vector 𝐁 and 𝐂. And so one way that we could answer this question is to begin by finding each of our cross products and then finding the dot products and then finding their sums.

But we can save ourselves a little bit of time. Suppose we have vector 𝐀 given as 𝐴 sub 𝑥, 𝐴 sub 𝑦, 𝐴 sub 𝑧; vector 𝐁, 𝐵 sub 𝑥, 𝐵 sub 𝑦, 𝐵 sub 𝑧; and vector 𝐂, 𝐶 sub 𝑥, 𝐶 sub 𝑦, 𝐶 sub 𝑧. The scalar triple product, which doesn’t actually need the parentheses, is given by the discriminant of the vector containing 𝐴 sub 𝑥, 𝐴 sub 𝑦, 𝐴 sub 𝑧, 𝐵 sub 𝑥, 𝐵 sub 𝑦, 𝐵 sub 𝑧, and 𝐶 sub 𝑥, 𝐶 sub 𝑦, 𝐶 sub 𝑧. So we can find an expression for the scalar triple product of 𝐀, 𝐁, and 𝐂. But what about 𝐁, 𝐂, and 𝐀 and 𝐂, 𝐀, and 𝐁?

Well, when we switch two rows in a three-by-three determinant, we know that changes the sign of the determinant. But then, if we switch two further rows, the sign changes back. And so what this means for scalar triple products is that they are equal if the cyclic order of the vectors is unchanged. In other words, the scalar triple product of 𝐀, 𝐁, and 𝐂 is equal to the scalar triple product of 𝐁, 𝐂, and 𝐀, which is in turn equal to the scalar triple product of 𝐂, 𝐀, and 𝐁. So the sum in question must actually be equal to three times the scalar triple product of 𝐀, 𝐁, and 𝐂. This saves us a lot of time because it means we can simply find this scalar triple product and then multiply it by three to answer our question.

So the scalar triple product we’re interested in is the determinant of the three-by-three matrix with elements one, two, four, two, four, negative one, and negative one, four, two. So next, we remind ourselves of how to find the determinant of a three-by-three matrix. We take each element on the top row. And we multiply it by the determinant of the two-by-two matrix that remains when we eliminate that row and that column. So to begin, we multiply one by the product of four and two minus the product of negative one and four. Then we subtract two multiplied by the product of two and two minus the product of negative one and negative one.

Finally, we add four multiplied by the product of two and four minus the product of four and negative one. Simplifying each part, and we get 12 minus six plus 48, which is equal to 54. So the scalar triple product of 𝐀, 𝐁, and 𝐂 is 54. According to our earlier calculations, we know that the answer to this question is three times this value, so it’s three times 54, which is equal to 162. The scalar triple product of 𝐀, 𝐁, and 𝐂 plus the scalar triple product of 𝐁, 𝐂, and 𝐀 plus the scalar triple product of 𝐂, 𝐀, and 𝐁 is 162.