3.16 The vector product

The scalar product of two vectors is a number. The vector product combines two vectors in three dimensions to give a third vector, which is perpendicular to both of them and whose length measures the area they span. These two facts make it the standard tool for finding a direction at right angles to a flat surface, such as the normal to a plane, and for calculating areas of shapes in space. In physics the same product gives the turning effect (torque) of a force.

Defining the vector product

The vector product is defined only for vectors in three dimensions. It is written with a cross, $\boldsymbol{v} \times \boldsymbol{w}$, and read "$\boldsymbol{v}$ cross $\boldsymbol{w}$", so it is also called the cross product.

The vector product formula is in the formula booklet, but it is quicker to calculate with a layout than to substitute into it.

Calculating a vector product by hand

Each component of $\boldsymbol{v} \times \boldsymbol{w}$ is a "cross-multiplication" of two rows of the column vectors, leaving out one row each time:

• First component: cover the first row, then multiply diagonally in the second and third rows: $v_2w_3 - v_3w_2$

In each case the product going "down to the right" ($v$ above, $w$ below) comes first and the one going "up to the right" is subtracted. The pattern $(2,3), (3,1), (1,2)$ of rows is cyclic, which is the easiest way to remember it.

Some textbooks write the same calculation as a determinant:

The minus sign in front of the $\boldsymbol{j}$ term reverses the order of the subtraction, so it gives the same second component as the definition.

Why the product is perpendicular to both vectors

The perpendicular property follows directly from the definition. Taking the scalar product of $\boldsymbol{v}$ with $\boldsymbol{v} \times \boldsymbol{w}$:

Every term cancels with another, so $\boldsymbol{v} \times \boldsymbol{w}$ is perpendicular to $\boldsymbol{v}$. The same calculation with $\boldsymbol{w}$ shows it is perpendicular to $\boldsymbol{w}$ as well.

Magnitude and direction

A vector is fixed by its magnitude and its direction. For the vector product, both have a geometric meaning.

Magnitude

If $\theta$ is the angle between $\boldsymbol{v}$ and $\boldsymbol{w}$ (placed tail to tail, $0 \le \theta \le \pi$), then

This formula is in the formula booklet. Since $0 \le \theta \le \pi$, $\sin\theta \ge 0$, so the right side is never negative. The magnitude is largest when the vectors are perpendicular ($\sin\theta = 1$) and zero when they are parallel ($\sin\theta = 0$

The result can be checked with the identity $\sin^2\theta + \cos^2\theta = 1$, as the next example shows. It also means that the scalar and vector products together contain all the information about the angle: the scalar product gives $\cos\theta$, the vector product gives $\sin\theta$

Direction and the right-hand rule

There are two directions perpendicular to both $\boldsymbol{v}$ and $\boldsymbol{w}$, opposite to each other. The right-hand rule picks out the one that $\boldsymbol{v} \times \boldsymbol{w}$ points along.

An equivalent version: point the index finger along $\boldsymbol{v}$ and the middle finger along $\boldsymbol{w}$; the thumb, held at right angles to both, points along $\boldsymbol{v} \times \boldsymbol{w}$. The rule matches the component formula because the coordinate axes are set up as a right-handed system, with $\boldsymbol{i} \times \boldsymbol{j} = \boldsymbol{k}$

Swapping the order reverses the turn, so the thumb points the other way. This is why the vector product depends on the order of the vectors, unlike the scalar product.

Products of the base vectors

The base vectors $\boldsymbol{i}$, $\boldsymbol{j}$ and $\boldsymbol{k}$ are unit vectors at right angles to each other, so the product of two different ones has magnitude $1 \times 1 \times \sin 90° = 1$

These six results, with $\boldsymbol{i} \times \boldsymbol{i} = \boldsymbol{j} \times \boldsymbol{j} = \boldsymbol{k} \times \boldsymbol{k} = \boldsymbol{0}$, are enough to expand any vector product written in $\boldsymbol{i}, \boldsymbol{j}, \boldsymbol{k}$

The magnitude formula and the scalar product can be combined to find $\sin\theta$ and $\cos\theta$ for the same pair of vectors, which shows what each product determines about an angle and what it leaves undecided.

Properties of the vector product

The vector product follows some, but not all, of the familiar rules of multiplication. Each property in the table can be checked from the component definition or from the geometric description.

Anticommutativity and a vector with itself

Swapping $\boldsymbol{v}$ and $\boldsymbol{w}$ in each component of the definition swaps the two terms of each subtraction, so every component changes sign: $\boldsymbol{w} \times \boldsymbol{v} = -(\boldsymbol{v} \times \boldsymbol{w})$

Associativity fails

Brackets matter in a repeated vector product. With base vectors,

so the two bracketings give different answers.

Testing for parallel vectors

Two non-zero vectors are parallel exactly when the angle between them is $0$ or $\pi$, which is exactly when $\sin\theta = 0$. So $\boldsymbol{v} \times \boldsymbol{w} = \boldsymbol{0}$

Finding a vector perpendicular to two vectors

Because $\boldsymbol{v} \times \boldsymbol{w}$ is perpendicular to both $\boldsymbol{v}$ and $\boldsymbol{w}$, the vector product is the direct way to find a direction at right angles to two given directions. This is how a normal vector to a plane is found from two directions in the plane (3.17), and how the direction of the common perpendicular of two skew lines is found (3.15). Any non-zero multiple of $\boldsymbol{v} \times \boldsymbol{w}$

Areas of parallelograms and triangles

A parallelogram with adjacent sides $\boldsymbol{v}$ and $\boldsymbol{w}$ has base $|\boldsymbol{v}|$ and perpendicular height $|\boldsymbol{w}|\sin\theta$. Its area is base times height, $|\boldsymbol{v}||\boldsymbol{w}|\sin\theta$

The parallelogram formula is in the formula booklet. The triangle formula follows because a diagonal splits the parallelogram into two congruent triangles; it is the vector version of the familiar $\frac{1}{2}ab\sin C$.

For a triangle $ABC$, the two vectors must start from the same vertex: use $\overrightarrow{AB}$ and $\overrightarrow{AC}$

When an area is given and a coordinate is unknown, the same formula leads to an equation for the unknown, usually a quadratic.

A modelling application: a shade sail

Shade sails are triangles of fabric stretched between three posts of different heights, so the sail is a tilted triangle in space. Its area decides how much fabric is needed.

A physics application: the moment of a force

In mechanics, a force $\boldsymbol{F}$ applied at a point with position vector $\boldsymbol{r}$ relative to a pivot has a turning effect, called its moment or torque, equal to $\boldsymbol{r} \times \boldsymbol{F}$. Its magnitude $|\boldsymbol{r}||\boldsymbol{F}|\sin\theta$

Exam-style questions on the vector product

Questions on the vector product usually chain several skills: find two vectors, find their product, then use it for an area, a perpendicular vector or a plane.

The vector product formula and $|\boldsymbol{v} \times \boldsymbol{w}| = |\boldsymbol{v}||\boldsymbol{w}|\sin\theta$ are both in the formula booklet, along with the area of a parallelogram; the triangle formula is not, so write $\frac{1}{2}|\overrightarrow{AB} \times \overrightarrow{AC}|$

Self-check

• To do:

  I can calculate $\boldsymbol{v} \times \boldsymbol{w}$ from components without sign errors, and check it with scalar products.

• To do:

  I can state the magnitude $|\boldsymbol{v}||\boldsymbol{w}|\sin\theta$ and use the right-hand rule to find the direction of a vector product.

• To do:

  I can work out products of $\boldsymbol{i}$