The scalar triple product, volumes and coplanarity

What you will be able to do

• find the scalar triple product, and the volume of a parallelepiped and of a tetrahedron;
• test whether three vectors, or four points, lie in one plane;
• use its rules, and the vector triple product.

A box that leans

A brick is a box with right angles, and its volume is length times breadth times height. A box can lean, like a pile of books pushed sideways. Its volume stays the same as long as the base and the height stay the same. Three vectors from one corner describe such a box, and one number gives its volume.

The scalar triple product

For three vectors, the scalar triple product is

It is a number. Other books write it $(\vec a, \vec b, \vec c)$ or $\vec a\cdot\vec b\times\vec c$. The NCERT chapter on vector algebra stops at the vector product. The scalar triple product is part of JEE Main, A-Level Further Maths and IB Higher Level.

Volume

Let the box have the edges $\vec a$, $\vec b$, $\vec c$ from one corner. The base on $\vec b$ and $\vec c$ has the area $\lvert\vec b\times\vec c\rvert$, and the vector $\vec b\times\vec c$ stands at right angles to the base. The height is the part of $\vec a$ along that direction, so $h = \lvert\vec a\cdot\hat n\rvert$. Then

The sign of $[\vec a\ \vec b\ \vec c]$ is positive when $\vec a$, $\vec b$, $\vec c$ form a right-handed set, like $\hat i$, $\hat j$, $\hat k$.

The rules

• Cyclic order. $[\vec a\ \vec b\ \vec c] = [\vec b\ \vec c\ \vec a] = [\vec c\ \vec a\ \vec b]$.
• Swapping two vectors changes the sign: $[\vec b\ \vec a\ \vec c] = -[\vec a\ \vec b\ \vec c]$.
• The dot and the cross may trade places: $\vec a\cdot(\vec b\times\vec c) = (\vec a\times\vec b)\cdot\vec c$.
• Zero: $[\vec a\ \vec b\ \vec c] = 0$ if two of the vectors are equal or parallel. More generally, it is zero exactly when the three vectors lie in one plane. Such vectors are called coplanar.

Coplanar points

Four points $A$, $B$, $C$, $D$ lie in one plane exactly when

Three vectors are coplanar when one of them is a combination of the other two, for example $\vec c = m\vec a + n\vec b$.

The vector triple product

The product $\vec a\times(\vec b\times\vec c)$ is a vector in the plane of $\vec b$ and $\vec c$. The rule is

Remember it as "the middle vector times the dot of the other two, minus the last". It differs from $(\vec a\times\vec b)\times\vec c$.

A worked example

Let $\vec a = 2\hat i + \hat k$, $\vec b = \hat i + 3\hat j - \hat k$ and $\vec c = \hat j + 4\hat k$.

1. The determinant. $

= 2(12 + 1) - 0 + 1(1 - 0) = 27$.

2. The volumes. The parallelepiped has volume $27$, and the tetrahedron has $\dfrac{27}{6} = 4.5$.
3. A coplanar triple. Take $\vec p = 2\vec b - \vec a$. Then $[\vec a\ \vec b\ \vec p] = 0$, because $\vec p$ is a combination of $\vec a$ and $\vec b$.
4. A vector triple product. With $\vec x = (1, 2, 0)$, $\vec y = (0, 1, 1)$ and $\vec z = (1, 0, 1)$, we have $\vec y\times\vec z = (1, 1, -1)$ and $\vec x\times(\vec y\times\vec z) = (-2, 1, -1)$. The rule gives $\vec y(\vec x\cdot\vec z) - \vec z(\vec x\cdot\vec y) = (0, 1, 1) - 2(1, 0, 1) = (-2, 1, -1)$.

Practise with new numbers

Try it in Python

The program finds the triple product of the worked example, the two volumes and a coplanar test.

Common mistakes

• Giving a negative volume. The triple product can be negative, but a volume is its absolute value.
• Forgetting the $\dfrac{1}{6}$ for a tetrahedron.
• Using the wrong order in a determinant. Each row is one vector, in order.
• Thinking the triple product is a vector. It is a number.
• Mixing the two triple products. $\vec a\cdot(\vec b\times\vec c)$ is a number, and $\vec a\times(\vec b\times\vec c)$ is a vector.

Remember these

• $[\vec a\ \vec b\ \vec c] = \vec a\cdot(\vec b\times\vec c)$, the determinant of the three rows.
• Parallelepiped: $\lvert[\vec a\ \vec b\ \vec c]\rvert$. Tetrahedron: $\dfrac{1}{6}\lvert[\vec a\ \vec b\ \vec c]\rvert$.
• Coplanar if and only if $[\vec a\ \vec b\ \vec c] = 0$.
• $\vec a\times(\vec b\times\vec c) = \vec b(\vec a\cdot\vec c) - \vec c(\vec a\cdot\vec b)$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find $[\hat i\ \hat j\ \hat k]$ and $[\hat j\ \hat i\ \hat k]$.

2. Find the volume of the parallelepiped with the edges $\hat i + \hat j$, $\hat j + \hat k$ and $\hat i + \hat k$.

= 1(1) - 1(0 - 1) + 0 = 2$.

3. Find the volume of the tetrahedron with the vertices $(0, 0, 0)$, $(2, 0, 0)$, $(0, 3, 0)$ and $(0, 0, 6)$.

4. Are $(1, 2, 3)$, $(2, 3, 4)$ and $(3, 5, 7)$ coplanar as vectors?

5. If $[\vec a\ \vec b\ \vec c] = 5$, what is $[\vec b\ \vec a\ \vec c]$?

References

• OpenStax, Calculus Volume 3, Chapter 2, section on the cross product, where the triple product and the volume of a parallelepiped appear.
• NCERT, Mathematics Part II for Class XII, Chapter 10, section on the vector product, for the cross product on which the triple product is built.