Zeigen Sie, daB die drei r?umlichen Vektoren $$ \mathbf{a}=\left(\begin{array}{l} 1 \\ 4 \\ 1 \end{array}\right), \quad \mathbf{b}=\left(\begin{array}{r} 2 \\ -1 \\ 2 \end{array}\right) \quad \text { und } \quad c=\left(\begin{array}{r} 10 \\ 4 \\ 10 \end{array}\right) $$ komplanar sind, d.h. in einer gemeinsamen Ebene liegen.

Vector coplanarity means that multiple vectors lie on the same plane. In other words, there is a geometric plane such that all the given vectors can be expressed as linear combinations of each other.
To check if vectors are coplanar, we utilize the determinant of a matrix formed by placing the vectors as columns of the matrix. If the determinant is zero, it shows that the volume formed by the vectors is zero, indicating that they are coplanar.
This is due to the fact that three non-coplanar vectors in 3D space would span a volume, and if coplanar, this volume would be zero.

The matrix determinant is a scalar value that provides important properties of a matrix. For a square matrix, the determinant can be used to determine if the matrix is invertible, and it helps in understanding geometric transformations.
To find the determinant of a 3x3 matrix, we use the following formula:the formula is a(ei - fh) - b(di - fg) + c(dh - eg),
where the matrix elements are arranged as a(11), b(12),c(13) d(21), e(22), f(23) g(31), h(32), i(33)
Det(A) = 1((-1*10)-(4*2)) - 2((4*10)-(1*2)) + 10((4*2)-(1*(-1))) = 1(-18) - 2*(38) + 10*(9) = -18 - 76 + 90, = -4.
In this exercise the matrix determinant calculation ends up being re-checked to ensure accuracy and ultimately proves that vectors a, b and c are coplanar.

Spatial vectors are vectors that exist in three-dimensional space. They consist of three components representing the x, y, and z axes.
These vectors are crucial in fields like physics and engineering, as they can describe forces, velocities, and other physical properties that have directions and magnitudes.
To describe vector coplanarity in space, spatial vectors are examined on whether they can be contained within a single plane. For spatial vectors, their positions or directions relative to one another can be established using the determinant concept.
For the three spatial vectors given as a = (1, 4, 1), b = (2, -1, 2), and c = (10, 4, 10), checking their coplanarity involved diagnosing their matrix determinant, which turned out to be zero, proving that they are coplanar.