The Scalar Triple Product

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Three vectors from one corner build a slanted box, and we will find its volume with one determinant. But in one course example, that determinant comes out as minus seven. Minus seven, as a volume? That sounds impossible, right? Soon you will know what that minus sign means. The same number also tells us when three vectors lie in one plane, and that comes up often in JEE questions. So, let's begin.

0:32Three vectors, one product

So, take three vectors a, b and c; the little arrow on top marks a vector. First b cross c, with the cross sign, inside the round brackets; then dot vector a with it, using the dot. We write the result as a b c in square brackets. So, is this a vector, or a number? It is a number, yes, exactly! The dot product of two vectors is always a scalar. That is why we call it the scalar triple product: three vectors in, one number out. Okay? Now in components. We write vector a as, a one times i hat, plus a two times j hat, plus a three times k hat; the small numbers written low are subscripts, and the hats mark unit vectors. The same way for b and c. Clear? Then the scalar triple product is just this determinant, the grid between the vertical lines. Row one is vector a, row two is vector b, and row three is vector c, in that order. Got it?

1:38What it means: the volume of a box

Now look at the figure. Vectors a, b and c start from the corner O, and build a slanted box called a parallelepiped. Its base is the shaded parallelogram on b and c, and its area is the modulus of b cross c; the vertical bars mean the modulus, the length. Right? The orange arrow is b cross c, perpendicular to the base. So the height, the dashed green line h, is the length of the projection of vector a on b cross c. Okay? Volume is base times height: the modulus of b cross c, times h, the height. As the figure shows, that is the modulus of vector a dot b cross c, the modulus of the scalar triple product. Clear? But our puzzle said minus seven, and a modulus is never negative. So what does the sign tell us? It tells us the order. The scalar triple product is greater than zero, positive, when a, b and c, in this order, form a right-handed system. So a minus sign is about the order, never a negative box. With me?

2:45Properties: order, sign and zero

Now the properties. Move the vectors round in a cycle, a b c, then b c a, then c a b, and the value stays the same. But what if we swap only two of them, say a and b? The sign changes, that is correct! Same box, same volume; only the sign flips. Keep this in mind for our puzzle. Okay? Next, the dot and the cross may be interchanged: vector a dot, b cross c, equals, a cross b, dot c. Keep the order a, b, c, and just swap the two symbols. Right? Now a quick one for you: what if the same vector comes twice, like a, then a again, then c, in square brackets? It is zero, yes, exactly! Swapping the two copies of a flips the sign but gives the same value, so the value is its own negative, which is zero. Two parallel vectors also give zero, because the box is flat. Got it?

3:46Coplanarity and volumes

This gives us the most useful test. Three vectors are coplanar, lying in one plane, if and only if their scalar triple product is zero; the double headed arrow means if and only if. Flat box, zero volume, and the other way round. Clear? For four points A, B, C and D, take three vectors from the first point: A B, A C and A D; an arrow over two letters runs from the first point to the second. The points are coplanar exactly when this product is zero. Okay? Now the volumes. The volume of the parallelepiped, our box, is the modulus of the triple product. The tetrahedron ABCD is a pyramid on a triangle, built on the same three edges. What fraction of the box is it? It is one sixth, yes! Half, since its base is a triangle, and a third, since it is a pyramid. So its volume is one sixth of the modulus of A B, A C, A D in square brackets. With me?

4:52Topper's trick: combinations of a, b and c

Now a trick that toppers use. When all three vectors are combinations of a, b and c, do not expand. Write the coefficients as a determinant, and multiply by a b c in square brackets; the cross sign here just means times. Okay? For example, take the vectors a plus b, b plus c, and c plus a. The coefficient rows are one, one, zero; zero, one, one; and one, zero, one. That determinant is two, so the answer is two times a b c, in just one line. Clear?

5:28Example 1: triple product is minus 7, so the volume is 7

Now a course example. Find the scalar triple product for vector a equal to two i hat minus three j hat plus four k hat, vector b equal to i hat plus two j hat minus k hat, and vector c equal to three i hat minus j hat plus two k hat, and the volume of the parallelepiped on them. Pause the video and try it first. Done? Okay, let's check. The rows are two, minus three, four; one, two, minus one; and three, minus one, two. Expand along the first row; the middle term carries a minus sign, and its entry is minus three. So, is it plus three, or minus three? It is plus three, because minus times minus is plus. So, two times, four minus one; plus three times, two plus three; plus four times, minus one minus six. Okay? That is six, plus fifteen, minus twenty eight, which gives minus seven. So the volume is minus seven, right? No, no, wait, this is our puzzle! The volume is the modulus of minus seven, so it is seven. The minus sign only says that vectors a, b, c in this order are not right-handed; swap any two, and you get plus seven, so our puzzle is solved. Clear?

6:52Example 2: volume of tetrahedron ABCD from four points

Now a harder one. Find the volume of the tetrahedron with these four corners; the numbers in round brackets are coordinates. Point A is one, two, three; B is two, three, one; C is three, one, two; and D is one, one, one. Here is a hint: use the edges that start at point A, okay? Pause the video and try it. Done? Okay, let's check. Subtract the first point from the others. So A B is one, one, minus two; A C is two, minus one, minus one; and A D is zero, minus one, minus two. Right? Put these three as the rows of a determinant, in this order: that is their scalar triple product. Clear? Expand along the first row: one times, two minus one; minus one times, minus four minus zero; minus two times, minus two minus zero. That gives one, plus four, plus four, which is nine. So the volume is nine, right? Oh no, careful, this is the trap! Nine is the whole box, and the tetrahedron is only one sixth of it. So one sixth times nine, that is nine over six, gives three over two. See how that one sixth saves you?

8:13Quick revision: determinant, volume, sign, coplanar test

So, quick revision. The scalar triple product, vector a dot b cross c, is the determinant of the components. Its modulus is the volume of the box, and a swap of two vectors only flips the sign. Zero means coplanar, and a tetrahedron is one sixth of the box. You did really well today. Now try the practice problems on this page. All the best!