Editing Euclidean vector (section)

===Scalar triple product===
{{main|Triple product#Scalar triple product|l1=Scalar triple product}}
The ''scalar triple product'' (also called the ''box product'' or ''mixed triple product'') is not really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is sometimes denoted by ('''a''' '''b''' '''c''') and defined as:

<math display=block>(\mathbf{a}\ \mathbf{b}\ \mathbf{c})
=\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}).</math>

It has three primary uses. First, the absolute value of the box product is the volume of the [[parallelepiped]] which has edges that are defined by the three vectors. Second, the scalar triple product is zero if and only if the three vectors are [[linear independence|linearly dependent]], which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectors '''a''', '''b''' and '''c''' are right-handed.

In components (''with respect to a right-handed orthonormal basis''), if the three vectors are thought of as rows (or columns, but in the same order), the scalar triple product is simply the [[determinant]] of the 3-by-3 [[Matrix (mathematics)|matrix]] having the three vectors as rows
<math display=block>(\mathbf{a}\ \mathbf{b}\ \mathbf{c})=\begin{vmatrix}
 a_1 & a_2 & a_3 \\
 b_1 & b_2 & b_3 \\
 c_1 & c_2 & c_3 \\
\end{vmatrix}</math>

The scalar triple product is linear in all three entries and anti-symmetric in the following sense:
<math display=block>
(\mathbf{a}\ \mathbf{b}\ \mathbf{c}) = (\mathbf{c}\ \mathbf{a}\ \mathbf{b}) = (\mathbf{b}\ \mathbf{c}\ \mathbf{a})=
 -(\mathbf{a}\ \mathbf{c}\ \mathbf{b}) = -(\mathbf{b}\ \mathbf{a}\ \mathbf{c}) = -(\mathbf{c}\ \mathbf{b}\ \mathbf{a}).</math>