Cross product in 3 dimensions

Vector calculus in plane and space: The Cross Product

Cross product in 3 dimensions

As we know, the dot product of two vectors is a number, as is the cross product in two dimensions. In three dimensions, there is also a cross product. It gives a vector with special properties when there are two vectors in the space. This involves a vector perpendicular to both given vectors, but an orientation is also required: a rule that tells you where the cross product is pointing. We shall determine an orientation first.

Right-handed orientation

Three vectors $\vec{v}$ , $\vec{w}$ , $\vec{u}$ (in this order), which are a basis of the space, are called right-handed oriented if $\vec{u}$ is on the side of the plane through $\vec{0}$ , $\vec{v}$ , $\vec{w}$ designated as follows by the middle finger: if $\vec{v}$ can be found along the extended thumb, and $\vec{w}$ lies along the extended index finger of the right hand, then the vector $\vec{u}$ points toward the outstretched middle finger.

The figure below represents a right-handed oriented basis.

The way to see whether a basis is right-hand-oriented, is also known as the corkscrew rule.

If a basis is not right-hand-oriented, then there is only one other possibility: it is left-hand-oriented. If $\vec{v}$ , $\vec{w}$ , $\vec{u}$ is right-hand-oriented, then $\vec{w}$ , $\vec{v}$ , $\vec{u}$ is left-hand-oriented.

The cross product

The cross product $\vec{v} \times \vec{w}$ of the vectors $\vec{v}$ and $\vec{w}$ , is the vector

$\vec{v}\times\vec{w}=\parallel\vec{v}\parallel\cdot\parallel\vec{w}\parallel\cdot \sin(\varphi)\cdot\vec{u}$ where

$\varphi$ is the short angle between $\vec{v}$ and $\vec{w}$ (so $0\leq \varphi \leq \pi$ ),
$\vec{u}$ is the vector of a length $1$ , perpendicular to $\vec{v}$ and $\vec{w}$ , so $\vec{v}$ , $\vec{w}$ , $\vec{u}$ is right-hand-oriented if $\varphi\ne0,\pi$ .

The vectors $\vec{v}$ and $\vec{w}$ are then and only then on a single line through the origin if $\varphi=0$ or $\pi$ applies. In both cases, $\sin(\varphi) = 0$ , so $\vec{v}\times\vec{w}=\vec{0}$ . In this case, the orientation of the vectors $\vec{v}$ , $\vec{w}$ , and $\vec{u}$ is not defined.

We choose $\vec{v}$ and $\vec{w}$ for the short angle. The oriented angle between $\vec{v}$ and $\vec{w}$ is actually undefined: the plane through the origin, $\vec{v}$ , and $\vec{w}$ does not, like any plane in space, necessarily have a top and bottom. In the two-dimensional theory, we see the plane from above, as it were. The oriented angle between the $x$ -axis and $y$ -axis, or rather between $\rv{1,0}$ and $\rv{0,1}$ , is measured by turning from one to the other counterclockwise. If we see these coordinates as the first two coordinates of the space, it corresponds to the right-handed basis $\rv{1,0,0}$ (the $x$ -axis), $\rv{0,1,0}$ (the $y$ -axis), and $\rv{0,0,1}$ (the $z$ -axis). Looking at the plane from above is described as looking from the position of third basis vector of a right-hand-oriented basis. In other words, only when we approach an orientation (such as 'counterclockwise' or ' $\vec{v}$ to $\vec{w}$ ') in the plane, can we speak of an oriented angle between two vectors. An oriented angle is not an obvious term, which explains why we avoid the oriented angle between $\vec{v}$ and $\vec{w}$ in the definition of the cross product in the space.

The scalar of the vector $\vec{u}$ in the definition of the cross product will never be negative, for $\sin(\varphi)\ge0$ because $\varphi$ describes the short angle between $\vec{v}$ and $\vec{w}$ .

The length of $\vec{v}\times\vec{w}$ is $\parallel\vec{v}\parallel\cdot\parallel\vec{w}\parallel\cdot\sin(\varphi)$ . If the vectors $\vec{v}$ and $\vec{w}$ , for example, both have a length $4$ , and the angle between the two vectors is equal to $60^{\circ}$ , then

$\vec{v}\times \vec{w} = 4\cdot 4\cdot \sin\left( 60^{\circ}\right) \cdot\vec{u}= 4\cdot 4\cdot \frac{1}{2}\sqrt{3}\cdot \vec{u}= 8\sqrt{3}\cdot\vec{u}$ where $\vec{u}$ is the unique vector of a length $1$ , so $\vec{v}$ , $\vec{w}$ , $\vec{u}$ is right-hand-oriented.

We come across other notations of the cross product in the literature, such as $\vec{v}\wedge\vec{w}$ .

The cross product is also called the vector product.

The cross product in 2 dimensions of the vectors $\vec{v}$ and $\vec{w}$ satisfies ${\vec{v}\times\vec{w} } = {\parallel\vec{v}\parallel\cdot \parallel\vec{w}\parallel}\cdot \sin(\varphi)$ . If we look at the plane as part of the space, then it is, except for a symbol, the scalar of the vector $\vec{u}$ perpendicular to that plane which appears in the definition of the cross product in three dimensions above.

The cross product in terms of two lengths

Let $\vec{v}$ and $\vec{w}$ be two vectors that are not together on a line through the origin. Use $\vec{u}$ to indicate the vector of a length $1$ that is perpendicular to $\vec{v}$ and $\vec{w}$ , so $\vec{v}$ , $\vec{w}$ , and $\vec{u}$ are right-hand-oriented. The cross product is then $\vec{v}\times\vec{w}$ equal to $\lambda\cdot\vec{u}$ , where $\lambda$ is the product of the length $\parallel\vec{v}\parallel$ of $\vec{v}$ and the distance from the endpoint of $\vec{w}$ to the line through $\vec{0}$ with directional vector $\vec{v}$ .

In particular, $\lambda$ is either equal to double the area of the triangle with vertices $\vec{0}$ , $\vec{v}$ , and $\vec{w}$ , or defined by the area of the parallelogram $\vec{v}$ and $\vec{w}$ (that is to say, with vertices $\vec{0}$ , $\vec{v}$ , $\vec{w}$ and $\vec{v}+\vec{w}$ ).

This follows directly from the definition of $\sin(\varphi)$ applied to the triangle $OAB$ ,where $O$ is the origin, $A$ is the end point of the representative of $\vec{w}$ with a starting point at the origin, and $B$ is the perpendicular projection of $A$ on the line through $O$ and the end point of the representative of $\vec{v}$ which is placed at the origin. See figure below.

The last statement follows from the length-times-height law for the surface of a parallelogram, in which the length is equal to $\parallel\vec{v}\parallel$ and the height is equal to $\parallel\vec{w}\parallel \cdot \sin( \varphi )$ .

Let $\vec{e}_1$ , $\vec{e}_2$ , $\vec{e}_3$ be a right-handed oriented orthonormal basis of the space. Determine $\vec{e}_{3}\times\vec{e}_{1}$ .

$\vec{e}_{3}\times \vec{e}_{1} = \vec{e}_{2}$

The formula for the cross product gives:

$\vec{e}_{3}\times\vec{e}_{1}=\parallel\vec{e}_{3}\parallel\cdot\parallel\vec{e}_{1}\parallel\cdot \sin(\varphi)\cdot\vec{u}$ which

$\varphi$ the (shortest) angle between $\vec{e}_{3}$ and $\vec{e}_{1}$ ,
$\vec{u}$ the vector of length $1$ is perpendicular to $\vec{e}_{3}$ and $\vec{e}_{1}$ , such that $\vec{e}_{3}$ , $\vec{e}_{1}$ , $\vec{u}$ is oriented right handed.

We can now determine the four factors of the right hand side of the formula as follows:

The lengths $\parallel\vec{e}_{3}\parallel$ and $\parallel\vec{e}_{1}\parallel$ are both equal to $1$ because the base is orthonormal.
The sine of the angle $\varphi$ is $1$ because $\vec{e}_{3}$ and $\vec{e}_{1}$ are perpendicular.
Vector $\vec{u}$ is $\vec{e}_{2}$ because the three vectors $\vec{e}_3$ , $\vec{e}_1$ , $\vec{e}_2$ are oriented right ${}$ handed .

The conclusion is that

$\vec{e}_{3}\times\vec{e}_{1}=1\cdot1\cdot 1 \cdot \vec{e}_{2} = \vec{e}_{2}\tiny.$

New example