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Cross product

The second kind of vector product is the so-called cross product or external product that results in a vector. It can be denoted in two alternative (and completely equivalent) ways,

(29)\[\begin{split}\vec{c}=\vec{a}\times\vec{b}\\ \vec{c}=\vec{a}\wedge\vec{b}\end{split}\]

In this course we will normally use the first one.

Definition

The cross product is defined as,

(30)\[\begin{split}\vec{c}=\vec{a}\times\vec{b}=\left\vert \begin{array}{ccc} \vec{i} & \vec{j} & \vec{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{array} \right\vert\end{split}\]

that can be express in components using the cofactor expansion of the determinant (minor expansion) in the following way:

(31)\[\vec{c}=\vec{a}\times\vec{b} = (a_yb_z-a_zb_y)\vec{i}+ (a_zb_x-a_xb_z)\vec{j}+ (a_xb_y-a_yb_x)\vec{k}\]

Geometric interpretation

The result of the cross product is a vector that (see Fig. Fig. 14):

  • is orthogonal to both initial vectors appearing in the product

  • its direction (sense) is given by the right-hand rule

  • its modulus is given in function of the modulii of \(\vec{a}\) and \(\vec{b}\) and the \(\sin\) of the angle they form (\(\theta\)):

(32)\[\vert \vec{a}\times\vec{b}\vert = \vert \vec{a}\vert \vert \vec{b}\vert \sin\theta\vec{c}\]
../_images/producto_vectorial.png

Fig. 14 Illustration of the cross product

Properties

Some properties of the cross product are:

  • Distributive: The crossproduct of a vector times a sum of vectors can be expanded in the following way,

(33)\[\vec{a}\times (\vec{b}+\vec{c}) = \vec{a}\times \vec{b}+\vec{a}\times\vec{c}\]

  • Product by a scalar: The result of multiplying a scalar times a cross product of vector is the same a multiplying only one of these vectors by the scalar

(34)\[\alpha(\vec{a}\times\vec{b})=(\alpha\vec{a})\times \vec{b}=\vec{a}\times(\alpha\vec{b})\]

  • Anticommutative: When the order of the vectors involved in the cross product is changed, the value of the result does not vary but its sign is reversed

(35)\[\vec{a}\times\vec{b}=-\vec{b}\times\vec{a}\]

  • Product of a vector by itself: The result of cross multiplying a vector by itself is the null vector

(36)\[\vec{a}\times\vec{a}=\vec{0}\]

Problems and solved examples

  • Show that the geometric properties of the cross product are correct

First we will show that the resulting vector of a cross product, \(\vec{a}\times\vec{b}\), is perpendicular to both \(\vec{a}\) and \(\vec{b}\).

Orthogonality between vectors

We know that two vectors are orthogonal if their scalar product is null

(37)\[\begin{split}\vec{a}\cdot(\vec{a}\times\vec{b})=\\(a_x\vec{i}+a_y\vec{j}+a_z\vec{k})\cdot \\ [(a_yb_z-a_zb_y)\vec{i}+(a_zb_x-a_xb_z)\vec{j}+(a_xb_y-a_yb_x)\vec{k}]=0\end{split}\]

This result is unchanged when using \(b\) instead of \(a\).

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We continue here

(38)\[y=-log(e)\]

And at the end

  • This is the text of another problem

Scalar product Mixed product