Chapter 4

Areas and Orientation

The dot product takes two vectors and produces a number which measures how strongly they point in the same direction. There is another natural question we can ask about a pair of vectors: how much area do they span?

Area alone will not quite tell the whole story. We also want to remember the order of the vectors---whether the turn from the first to the second is clockwise or counterclockwise in the plane, or which side of their plane is positive in space. This extra choice is called orientation.

4.1Signed Area and the Planar Determinant

We begin in the plane, where two vectors span a parallelogram. The formula which measures its signed area is the determinant.

Definition 4.1 (The $2\times2$ Determinant). The determinant of a $2 \times 2$ matrix is

∣ 𝑎 𝑏 𝑐 𝑑 ∣ = 𝑎 𝑑 - 𝑏 𝑐 .

For plane vectors $⃗ 𝑢 = ⟨ 𝑢 𝑥, 𝑢 𝑦 ⟩$ and $⃗ 𝑣 = ⟨ 𝑣 𝑥, 𝑣 𝑦 ⟩$ , we write

d e t (⃗ 𝑢, ⃗ 𝑣) = ∣ 𝑢 𝑥 𝑢 𝑦 𝑣 𝑥 𝑣 𝑦 ∣ = 𝑢 𝑥 𝑣 𝑦 - 𝑢 𝑦 𝑣 𝑥 .

The absolute value $| d e t (⃗ 𝑢, ⃗ 𝑣) |$ is the area of the parallelogram spanned by $⃗ 𝑢$ and $⃗ 𝑣$ . The sign records its orientation:

$d e t (⃗ 𝑢, ⃗ 𝑣) > 0$ when the turn from $⃗ 𝑢$ to $⃗ 𝑣$ is counterclockwise;
$d e t (⃗ 𝑢, ⃗ 𝑣) < 0$ when the turn is clockwise;
$d e t (⃗ 𝑢, ⃗ 𝑣) = 0$ when the vectors are parallel and the parallelogram has collapsed onto a line.

For example, if $⃗ 𝑢 = ⟨ 2, 1 ⟩$ and $⃗ 𝑣 = ⟨ 1, 3 ⟩$ , then

d e t (⃗ 𝑢, ⃗ 𝑣) = 2 (3) - 1 (1) = 5 .

Thus the parallelogram has area $5$ , and the turn from $⃗ 𝑢$ to $⃗ 𝑣$ is counterclockwise. Reversing the order reverses the sign:

d e t (⃗ 𝑣, ⃗ 𝑢) = - 5 .

The parallelogram has not changed, so its ordinary area is still $5$ ; only its orientation has changed.

4.2The Cross Product

In two dimensions, given one vector $⃗ 𝑣 = ⟨ 𝑎, 𝑏 ⟩$ , it is easy to find an orthogonal vector to it: $⃗ 𝑣 ⟂ = ⟨ 𝑏, - 𝑎 ⟩$ , for example.

In three dimensions, the analogous problem is more difficult: given two vectors, how do we find a third vector that is orthogonal to them both at once? A nice solution is given by the cross product.

There is an entire line of vectors orthogonal to both $⃗ 𝑢$ and $⃗ 𝑣$ , so orthogonality alone does not determine one of them. To specify the cross product, we need to describe its magnitude and its direction as well.

Definition 4.2 (Cross Product as an Oriented Area Vector). For vectors $⃗ 𝑢, ⃗ 𝑣 \in ℝ 3$ , the cross product $⃗ 𝑢 \times ⃗ 𝑣$ is characterized geometrically by three properties:

It is orthogonal to both $⃗ 𝑢$ and $⃗ 𝑣$ .
Its magnitude is

‖ ⃗ 𝑢 \times ⃗ 𝑣 ‖ = ‖ ⃗ 𝑢 ‖ ‖ ⃗ 𝑣 ‖ s i n 𝜃,

the area of the parallelogram spanned by $⃗ 𝑢$ and $⃗ 𝑣$ . 3. Its direction is chosen by the right-hand rule.

The formula “base times height,” $‖ ⃗ 𝑢 ‖ ‖ ⃗ 𝑣 ‖ s i n 𝜃$ , may be familiar as the area of a parallelogram. Thus the cross product packages both the area of the parallelogram and a direction perpendicular to it into one vector. It is an oriented area vector.

If $⃗ 𝑢$ and $⃗ 𝑣$ are parallel, the parallelogram they span collapses onto a line. Lines have zero area, so the cross product has zero length and must be the zero vector.

This gives us the magnitude information, but what about the direction? There are two possible directions perpendicular to the plane containing $⃗ 𝑢$ and $⃗ 𝑣$ .

Theorem 4.3 (The Right-Hand Rule). Align the fingers of your right hand with $⃗ 𝑢$ and curl them toward $⃗ 𝑣$ . Your thumb points in the direction of $⃗ 𝑢 \times ⃗ 𝑣$ .

Reversing the order reverses this direction, so

⃗ 𝑢 \times ⃗ 𝑣 = - ⃗ 𝑣 \times ⃗ 𝑢 .

The order of the vectors is therefore essential: the cross product is not commutative.

We can now state the exact relationship between the planar determinant and the cross product. Place two plane vectors in the $𝑥 𝑦$ -plane of $ℝ 3$ by adding a zero $𝑧$ -component. Their cross product must point along the $𝑧$ -axis, and its $𝑧$ -component is precisely their determinant:

⟨ 𝑢 𝑥, 𝑢 𝑦, 0 ⟩ \times ⟨ 𝑣 𝑥, 𝑣 𝑦, 0 ⟩ = ⟨ 0, 0, d e t (⃗ 𝑢, ⃗ 𝑣) ⟩ .

This does not make the planar determinant a “two-dimensional cross product.” The determinant is a scalar measuring signed area in $ℝ 2$ ; the cross product is a vector defined in $ℝ 3$ . The figure below shows how the two are related.

Figure 4.1 On the left, the magnitude of the determinant is area and its sign records the turn from

⃗ 𝑢

⃗ 𝑣

. On the right, the same vectors lie in the

𝑥 𝑦

-plane and their cross product carries that signed area along the

𝑧

-axis.

The same geometry continues when the vectors are allowed to point anywhere in three-dimensional space.

Figure 4.2 The cross product remains orthogonal to both input vectors, and its length is the area of their parallelogram. Swap the order to reverse its direction, or make the two vectors parallel to collapse both the area and the cross product to zero.

4.3Computing the Cross Product

The geometric description tells us what the cross product means. To compute it from components, we use the following formula.

Definition 4.4 (Cross Product in Coordinates). If $⃗ 𝑢 = ⟨ 𝑢 𝑥, 𝑢 𝑦, 𝑢 𝑧 ⟩$ and $⃗ 𝑣 = ⟨ 𝑣 𝑥, 𝑣 𝑦, 𝑣 𝑧 ⟩$ , then

⃗ 𝑢 \times ⃗ 𝑣 = ⟨ 𝑢 𝑦 𝑣 𝑧 - 𝑢 𝑧 𝑣 𝑦, 𝑢 𝑧 𝑣 𝑥 - 𝑢 𝑥 𝑣 𝑧, 𝑢 𝑥 𝑣 𝑦 - 𝑢 𝑦 𝑣 𝑥 ⟩ .

This formula looks complicated, but it is built out of the $2 \times 2$ determinants we already know:

⃗ 𝑢 \times ⃗ 𝑣 = ∣ 𝑢 𝑦 𝑢 𝑧 𝑣 𝑦 𝑣 𝑧 ∣ ˆ ı - ∣ 𝑢 𝑥 𝑢 𝑧 𝑣 𝑥 𝑣 𝑧 ∣ ˆ ȷ + ∣ 𝑢 𝑥 𝑢 𝑦 𝑣 𝑥 𝑣 𝑦 ∣ ˆ 𝑘 .

Note the minus sign on the second term. A common mnemonic packages this expansion into the determinant notation

⃗ 𝑢 \times ⃗ 𝑣 = ∣ ∣ ∣ ∣ ˆ ı ˆ ȷ ˆ 𝑘 𝑢 𝑥 𝑢 𝑦 𝑢 𝑧 𝑣 𝑥 𝑣 𝑦 𝑣 𝑧 ∣ ∣ ∣ ∣ .

Definition 4.5 (The $3\times3$ Determinant). A $3 \times 3$ determinant can be expanded along its first row using the $2 \times 2$ determinants defined above:

∣ 𝑥 𝑦 𝑧 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 ∣ = 𝑥 ∣ 𝑏 𝑐 𝑒 𝑓 ∣ - 𝑦 ∣ 𝑎 𝑐 𝑑 𝑓 ∣ + 𝑧 ∣ 𝑎 𝑏 𝑑 𝑒 ∣ .

For example,

⟨ 1, 2, 0 ⟩ \times ⟨ 3, - 1, 4 ⟩ = ⟨ 2 (4) - 0 (- 1), 0 (3) - 1 (4), 1 (- 1) - 2 (3) ⟩ = ⟨ 8, - 4, - 7 ⟩ .

Exercise 4.6 (Orthogonality of the Cross Product). Check that $⟨ 8, - 4, - 7 ⟩$ is orthogonal to both $⟨ 1, 2, 0 ⟩$ and $⟨ 3, - 1, 4 ⟩$ .

Then check directly from the coordinate formula that if $⃗ 𝑢 = ⟨ 𝑎, 𝑏, 𝑐 ⟩$ and $⃗ 𝑣 = ⟨ 𝑥, 𝑦, 𝑧 ⟩$ , the vector $⃗ 𝑢 \times ⃗ 𝑣$ is orthogonal to both $⃗ 𝑢$ and $⃗ 𝑣$ .

The cross product distributes across addition and scalar multiplication:

⃗ 𝑢 \times (⃗ 𝑣 + ⃗ 𝑤) = ⃗ 𝑢 \times ⃗ 𝑣 + ⃗ 𝑢 \times ⃗ 𝑤,

(𝑐 ⃗ 𝑢) \times ⃗ 𝑣 = 𝑐 (⃗ 𝑢 \times ⃗ 𝑣) = ⃗ 𝑢 \times (𝑐 ⃗ 𝑣) .

But it is not ordinary multiplication. We have already seen that reversing the order changes the sign. It is also not associative: in general,

(⃗ 𝑢 \times ⃗ 𝑣) \times ⃗ 𝑤 \neq ⃗ 𝑢 \times (⃗ 𝑣 \times ⃗ 𝑤) .

When several cross products occur in one expression, the parentheses matter.

It is often useful to know the cross products of the standard basis vectors:

ˆ ı \times ˆ ȷ = ˆ 𝑘, ˆ ȷ \times ˆ 𝑘 = ˆ ı, ˆ 𝑘 \times ˆ ı = ˆ ȷ .

Reading these products backward inserts a minus sign; for example, $ˆ ȷ \times ˆ ı = - ˆ 𝑘$ .

4.4Oriented Volume and the Scalar Triple Product

If we have three vectors, it is possible to combine the dot and cross products to get a single number: take the cross product of two of them to get an oriented area vector, then dot that result with the third.

Definition 4.7 (Scalar Triple Product). The scalar triple product of vectors $⃗ 𝑢, ⃗ 𝑣$ , and $⃗ 𝑤$ is

⃗ 𝑢 \cdot (⃗ 𝑣 \times ⃗ 𝑤) .

Equivalently, it is the $3 \times 3$ determinant

⃗ 𝑢 \cdot (⃗ 𝑣 \times ⃗ 𝑤) = d e t (⃗ 𝑢, ⃗ 𝑣, ⃗ 𝑤) = ∣ ∣ ∣ ∣ 𝑢 𝑥 𝑢 𝑦 𝑢 𝑧 𝑣 𝑥 𝑣 𝑦 𝑣 𝑧 𝑤 𝑥 𝑤 𝑦 𝑤 𝑧 ∣ ∣ ∣ ∣ .

Just as the $2 \times 2$ determinant measures the signed area of a parallelogram, this $3 \times 3$ determinant measures the signed volume of a parallelepiped.

The cross product $⃗ 𝑣 \times ⃗ 𝑤$ gives the oriented area of the base. Dotting it with $⃗ 𝑢$ multiplies that base area by the signed height of $⃗ 𝑢$ above the base. Therefore

∣ ⃗ 𝑢 \cdot (⃗ 𝑣 \times ⃗ 𝑤) ∣

is the ordinary volume. The sign records whether the ordered set $(⃗ 𝑢, ⃗ 𝑣, ⃗ 𝑤)$ has positive or negative orientation. If the triple product is zero, the three vectors are coplanar and the box has collapsed.

For example, let

⃗ 𝑢 = ⟨ 1, 0, 1 ⟩, ⃗ 𝑣 = ⟨ 2, 1, 0 ⟩, ⃗ 𝑤 = ⟨ 0, 1, 2 ⟩ .

Then

⃗ 𝑣 \times ⃗ 𝑤 = ⟨ 2, - 4, 2 ⟩

and

⃗ 𝑢 \cdot (⃗ 𝑣 \times ⃗ 𝑤) = 4 .

Thus the parallelepiped has volume $4$ , and the ordered triple has positive orientation.

Figure 4.3 The base parallelogram on

⃗ 𝑣

and

⃗ 𝑤

has area

‖ ⃗ 𝑣 \times ⃗ 𝑤 ‖

. The scalar triple product multiplies this by the signed height of

⃗ 𝑢

. Its magnitude is volume, and its sign records orientation.

Determinants now give us one common language for signed area in the plane, oriented area vectors in space, and signed volume. In the next chapter, the cross product will let us construct a normal vector to the plane through three points, while the dot product will turn that normal direction into an equation for the plane.