Aus Band 1 , Abschnitt II.3.5 ist bekannt: Das Spatprodukt \([\vec{a} \vec{b} \vec{c}]=\vec{a} \cdot(\vec{b} \times \vec{c})\) dreier Vektoren aus dem Anschauungsraum kann aus den skalaren Vektorkomponenten mit Hilfe der Determinante $$ [\vec{a} \vec{b} \vec{c}]=\left|\begin{array}{lll} a_{x} & a_{y} & a_{z} \\ b_{x} & b_{y} & b_{z} \\ c_{x} & c_{y} & c_{z} \end{array}\right| $$ berechnet werden. Zeigen Sie, daB die drei Vektoren $$ \vec{a}=\left(\begin{array}{l} 1 \\ 2 \\ 2 \end{array}\right), \quad \vec{b}=\left(\begin{array}{r} 0 \\ -4 \\ 3 \end{array}\right) \text { und } \vec{c}=\left(\begin{array}{r} 3 \\ -6 \\ 15 \end{array}\right) $$ komplanar sind, d.h. in einer Ebene liegen.

The vector product, also known as the cross product, is an operation on two vectors in three-dimensional space. For two vectors \(\vec{u}\) and \(\vec{v}\), their cross product \(\vec{u} \times \vec{v}\) results in a third vector that is perpendicular to both. This vector product is essential in various fields, including physics and engineering, for finding directions perpendicular to two given lines. The magnitude of \(\vec{u} \times \vec{v}\) is given by \( |\vec{u}| |\vec{v}| \sin(\theta)\), where \(\theta\) is the angle between \(\vec{u}\) and \(\vec{v}\). To find the cross product, use the matrix determinant method involving the unit vectors \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\).

Vectors are said to be coplanar if they lie within the same plane. An important property of three vectors in three-dimensional space is that if the volume of the parallelepiped they form is zero, then these vectors are coplanar. This condition can be tested using the scalar triple product \[ (\vec{a} \cdot (\vec{b} \times \vec{c}) )\]. If the result is zero, then the vectors are coplanar.
Another way to check for coplanarity is to set up a determinant with the vectors as rows (or columns). If the determinant of this matrix equals zero, it indicates that the vectors do not span a three-dimensional space but instead lie in a two-dimensional plane.

The 3x3 determinant expansion is a crucial concept in linear algebra, used for calculations involving three-dimensional vectors. It is essential for solving systems of linear equations and finding the area or volume bounded by vectors.
We can find the determinant of a \(3x3\) matrix by expanding it along any row or column. For example, given the matrix \[\begin{pmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{pmatrix}\], the determinant can be computed as:
\[a_{11} \begin{vmatrix} a_{22} & a_{23} \ a_{32} & a_{33} \end{vmatrix} - a_{12} \begin{vmatrix} a_{21} & a_{23} \ a_{31} & a_{33} \end{vmatrix} + a_{13} \begin{vmatrix} a_{21} & a_{22} \ a_{31} & a_{32} \end{vmatrix} \] This method breaks the 3x3 determinant into smaller 2x2 determinants, making it easier to calculate.

In linear algebra, vectors are linearly dependent if there exists a set of coefficients (not all zero) such that a linear combination of these vectors equals the zero vector. Essentially, one vector can be written as a combination of the others.
For three vectors \(\vec{a}, \vec{b}, \vec{c}\), they are linearly dependent if the determinant of the matrix formed by these vectors is zero. This means they lie in the same plane or line and do not span a three-dimensional space. The concept of linear dependence is foundational in vector spaces, as it helps in understanding vector relations and performing vector space analyses.
If \(\vec{a}, \vec{b},\vec{c}\) are given as:\(\begin{pmatrix} 1 \ 2 \ 2 \end{pmatrix}, \begin{pmatrix} 0 \ -4 \ 3 \end{pmatrix}, \begin{pmatrix} 3 \ -6 \ 15 \end{pmatrix}\), checking the determinant helps us conclude that they are coplanar and hence linearly dependent.