C. Geometry and Algebra of Vectors · Dot and Cross Products

The Triple Scalar Product: Volume and Coplanar Vectors

C2.8About 65 minutes

Learning goals

I can explain why the volume of a parallelepiped is |a · (b × c)|.
I can calculate the volume of a parallelepiped from vectors or from vertices.
I can use the triple scalar product to test whether three vectors (or four points) are coplanar.
I can use the cyclic and sign properties of the triple scalar product.

Words to know

parallelepiped
A solid with six faces that are parallelograms, opposite faces parallel; a "slanted box" determined by three edge vectors from one vertex.

triple scalar product
a · (b × c), a number whose absolute value is the volume of the parallelepiped determined by a, b and c.

coplanar
Lying in one plane. Three vectors are coplanar exactly when a · (b × c) = 0.

tetrahedron
A pyramid with a triangular base; four triangular faces.

From area to volume

The dot product gives lengths and angles. The cross product gives areas. Put them together and you get volume.

Three vectors a, b and c starting at one point determine a slanted box called a parallelepiped (par-uh-lel-uh-PIE-ped): a solid whose six faces are parallelograms, with opposite faces parallel. A shoebox is a special case where every angle is 90°. A stack of paper pushed sideways, or a crystal of calcite, has this shape with slanted sides.

The volume of any prism is base area × height. For a parallelepiped:

The base is the parallelogram on b and c, so its area is |b × c| (lesson 8).
b × c is perpendicular to the base. The height h is the length of the part of a in that perpendicular direction: the absolute value of the scalar projection of a on b × c, which is |a · (b × c)| / |b × c| (lesson 5).

Multiply: V = |b × c| × |a · (b × c)| / |b × c|, so

V = |a · (b × c)|

The number a · (b × c) is called the triple scalar product (or scalar triple product). Notice the order of operations: the cross product first (it must be, since a · b is a number and you cannot cross a number with c).

A parallelepiped with edges a, b and c. The base is the parallelogram on b and c, with area |b × c|. The height h is the part of a along the normal b × c. Multiplying gives V = |a · (b × c)|.

Worked example

Example 1: A volume you can check

Find the volume of the parallelepiped determined by a = [2, 0, 0], b = [1, 3, 0] and c = [1, 1, 4]. Check it with base × height.

b × c = [(3)(4) − (0)(1), (0)(1) − (1)(4), (1)(1) − (3)(1)] = [12, −4, −2].
a · (b × c) = 2(12) + 0(−4) + 0(−2) = 24.
V = |24| = 24.
Check: a and b lie in the xy-plane, so they form a base of area |a × b| = |[0, 0, 6]| = 6. The third edge c rises 4 units (its z-component), so the height is 4. Volume 6 × 4 = 24 ✓.

V = 24 cubic units

Worked example

Example 2: A negative triple product

Find the volume of the parallelepiped determined by a = [1, 2, −1], b = [3, 0, 2] and c = [−1, 4, 1].

b × c = [(0)(1) − (2)(4), (2)(−1) − (3)(1), (3)(4) − (0)(−1)] = [−8, −5, 12].
a · (b × c) = −8 − 10 − 12 = −30.
V = |−30| = 30.

V = 30 cubic units. The negative sign only means a is on the opposite side of the base from b × c (the three vectors, in this order, form a left-handed set). Volume is always the absolute value.

Properties of the triple scalar product

Cyclic order does not change it: a · (b × c) = b · (c × a) = c · (a × b). Any of the three faces can be the base; the box is the same.
Swapping two vectors changes the sign: a · (c × b) = −a · (b × c), because c × b = −(b × c).
Dot and cross can trade places: a · (b × c) = (a × b) · c.
If two of the vectors are equal or parallel, it is 0. For example a · (a × b) = 0, which you proved in lesson 9.
For the unit vectors: i · (j × k) = i · i = 1, the volume of the unit cube.

When three vectors lie in one plane (here w = u + v), the "box" they make is flat, so its volume is 0. A triple scalar product of 0 is a test for coplanar vectors.

Coplanar vectors

Three vectors are coplanar if they lie in one plane when placed tail to tail. Then the parallelepiped is flat and has no volume. So:

a · (b × c) = 0 if and only if a, b and c are coplanar.

This also tests whether four points lie in one plane: build three vectors from one of the points to the other three, and compute their triple scalar product.

Worked example

Example 3: A coplanarity test

Are u = [1, 2, 3], v = [4, 5, 6] and w = [7, 8, 9] coplanar?

v × w = [(5)(9) − (6)(8), (6)(7) − (4)(9), (4)(8) − (5)(7)] = [−3, 6, −3].
u · (v × w) = −3 + 12 − 9 = 0.

Yes, they are coplanar. (In fact w = 2v − u, so w is built from u and v and lies in their plane.)

Worked example

Example 4: Making vectors coplanar

Find k so that [1, 1, k], [2, 0, 1] and [0, 3, 1] are coplanar.

[2, 0, 1] × [0, 3, 1] = [(0)(1) − (1)(3), (1)(0) − (2)(1), (2)(3) − (0)(0)] = [−3, −2, 6].
[1, 1, k] · [−3, −2, 6] = −3 − 2 + 6k = 6k − 5.
Set 6k − 5 = 0: k = 5/6.

k = 5/6

Worked example

Example 5: Volume from vertices

A parallelepiped has a vertex P(1, 1, 1), and the three vertices joined to P by edges are Q(3, 1, 2), R(1, 4, 1) and S(2, 2, 5). Find its volume.

Edges from P: a = PQ = [2, 0, 1], b = PR = [0, 3, 0], c = PS = [1, 1, 4].
b × c = [(3)(4) − (0)(1), (0)(1) − (0)(4), (0)(1) − (3)(1)] = [12, 0, −3].
a · (b × c) = 24 + 0 − 3 = 21.

V = 21 cubic units

Volume in four steps

Make three edge vectors that start at the same vertex.
Cross two of them.
Dot the result with the third.
Take the absolute value for volume. A result of 0 means the vectors are coplanar.

Just for fun

"Parallelepiped" has a reputation as one of the hardest words to pronounce in high-school math. Practise: par-uh-lel-uh-PIE-ped. Or, like many students, just call it a squished box.

Hands-on activity

The sheared deck

You need: a deck of playing cards (or a thick pad of sticky notes, or a stack of 50 to 100 sheets of paper), a ruler and a calculator.

Stack the cards neatly. Measure the length ℓ, the width w and the thickness t of the stack in centimetres. The volume is ℓwt.
Push the top of the stack forward and sideways so it leans in two directions at once. Now it is a parallelepiped.
Set up axes: x along the card length, y along the card width, z up, with the origin at a bottom corner. The bottom edges are a = [ℓ, 0, 0] and b = [0, w, 0]. Measure how far the top card has moved forward (p) and sideways (q) compared with the bottom card. The slanted edge is c = [p, q, t].
Compute |c · (a × b)| and compare with ℓwt.

What to look for: a × b = [0, 0, ℓw], so c · (a × b) = ℓwt no matter what p and q are. The cards are the same cards, so the volume cannot change, and the formula agrees. For an 8.9 cm × 6.4 cm × 1.6 cm deck, the volume stays about 91 cm³.

Your turn

Find the volume of the parallelepiped determined by [1, 0, 0], [0, 2, 0] and [0, 0, 3].

Find the volume determined by a = [2, 1, 0], b = [0, 1, 3] and c = [1, 0, 1].

Find the volume determined by [1, 1, 0], [0, 1, 1] and [1, 0, 1].

Are [1, 2, 1], [2, 0, 3] and [3, 2, 4] coplanar?

a · (b × c) = 7. Find c · (a × b) and b · (a × c).

Use the triple scalar product to find the volume of a rectangular box 2 by 3 by 4.

More practice: warm-up

Write the formula for the volume of the parallelepiped determined by a, b and c.

a · (b × c) = 0. What does that tell you?

Find i · (j × k).

Find j · (i × k).

a · (b × c) = −15. What is the volume?

More practice: on your own

A parallelepiped has vertex A(0, 1, 2) with neighbouring vertices B(3, 1, 2), C(1, 4, 2) and D(1, 2, 5). Find its volume.

Find k so that [k, 1, 2], [1, −1, 0] and [2, 1, 1] are coplanar.

Do P(1, 0, 0), Q(0, 1, 0), R(0, 0, 1) and S(1, 1, −1) lie in one plane?

For a = [1, 2, −1], b = [3, 0, 2] and c = [−1, 4, 1] (Example 2), find the height of the parallelepiped above the base formed by b and c.

Explain, using volume, why a · (a × b) = 0.

Challenge

A tetrahedron (triangular pyramid) with edges a, b and c from one vertex has volume (1/6)|a · (b × c)|. Find the volume of the tetrahedron with vertices O(0, 0, 0), (1, 0, 0), (0, 2, 0) and (0, 0, 3).

Prove that a · (b × c) = (a × b) · c.

Find all real t for which the parallelepiped on [1, t, 0], [0, 1, t] and [t, 0, 1] has volume 9.

Check your understanding

Quiz

What is the volume of the parallelepiped determined by [2, 0, 0], [0, 3, 0] and [0, 0, 4]?

a · (b × c) = 0 for three non-zero vectors. What does this mean?

What is i · (j × k)?

a · (b × c) = −12. What is the volume of the parallelepiped?

a · (b × c) = 5. What is b · (c × a)?

Which expression has no meaning?

For parents and teachers

Big idea (C2.8): the triple scalar product a · (b × c) combines both products. Its absolute value is the volume of the parallelepiped (slanted box) with edges a, b and c, and it is 0 exactly when the three vectors are coplanar.

How to help: The sheared-deck activity shows why pushing a stack sideways does not change its volume, and the formula agrees. Ask your teen to explain why the cross product must be done first in a · (b × c).

Common difficulties:

Doing the dot product first, which leads to a meaningless "number cross vector".
Forgetting the absolute value for volume.
Using edge vectors that do not all start at the same vertex.
Sign errors inside the cross product; check b × c by dotting it with b and c before using it.