Question

Gegeben ist ein Dreieck \(\triangle \mathbf{0} A B\) mit den Punkten \(A=(1,4,2)\) und \(B=(2,5,1)\). 0 ist der Ursprung im \(\mathbb{R}^{3}\). Berechnen Sie mit den Mitteln der Vektorrechnung den Flächeninhalt des Dreiecks.

Short answer

The area of \( \triangle OAB \) is \( \frac{3\sqrt{6}}{2} \).

Step-by-step solution

Vector Representation of Triangle Sides

The triangle is formed by vectors \( \overrightarrow{OA} \), \( \overrightarrow{OB} \), and \( \overrightarrow{AB} \). First, express vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \). Since \( O \) is the origin, \( \overrightarrow{OA} = A - O = (1, 4, 2) \) and \( \overrightarrow{OB} = B - O = (2, 5, 1) \).

Find Cross Product of Vectors

To find the area of \( \triangle OAB \), calculate the magnitude of the cross product \( \overrightarrow{OA} \times \overrightarrow{OB} \). The cross product \( \overrightarrow{OA} \times \overrightarrow{OB} \) is calculated by taking the determinant of: \[\overrightarrow{OA} \times \overrightarrow{OB} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 4 & 2 \ 2 & 5 & 1 \end{vmatrix}\]= \( \mathbf{i}(4 \cdot 1 - 2 \cdot 5) - \mathbf{j}(1 \cdot 1 - 2 \cdot 2) + \mathbf{k}(1 \cdot 5 - 4 \cdot 2) \) = \(-6\mathbf{i} + 3\mathbf{j} - 3\mathbf{k}\).

Calculate Magnitude of Cross Product

Now, find the magnitude of the vector \( \overrightarrow{OA} \times \overrightarrow{OB} = (-6, 3, -3) \). Use the formula for the magnitude: \[||\overrightarrow{OA} \times \overrightarrow{OB}|| = \sqrt{(-6)^2 + 3^2 + (-3)^2} = \sqrt{36 + 9 + 9} = \sqrt{54} = 3\sqrt{6}.\]

Calculate Area of Triangle

The area of the triangle \( \triangle OAB \) is half of the magnitude of the cross product. Thus, the area is given by: \[\text{Area} = \frac{1}{2} \times ||\overrightarrow{OA} \times \overrightarrow{OB}|| = \frac{1}{2} \times 3\sqrt{6} = \frac{3\sqrt{6}}{2}.\]

Key concepts

Dreiecksfläche

The concept of "Dreiecksfläche," or the area of a triangle, can be understood by using vector calculus. In three-dimensional space, given a triangle with vertices at specific points, the challenge is to find the area. The given triangle has vertices at the origin and two other points, A and B, described by their coordinates.
To find the area of this triangle, one useful approach is applying the formula involving vectors. Triangles in space pose a slightly different challenge due to their three-dimensional nature. But with vectors, the area calculation becomes structured and formulaic.
By taking the cross product of two vectors lying in the plane of the triangle, we can find the vector perpendicular to the plane, then use its magnitude to calculate the area. The formula for the area becomes \[ \text{Area} = \frac{1}{2} \, \|\overrightarrow{u} \times \overrightarrow{v}\| \] where \( \overrightarrow{u} \) and \( \overrightarrow{v} \) are vectors that define the plane. This method is particularly useful for providing an accurate geometrical interpretation based on the vectors' properties.

Kreuzprodukt

The "Kreuzprodukt," or cross product, is a fundamental operation in vector algebra, especially when dealing with three-dimensional vectors. It is represented by the multiplication sign \( \times \). In our context of calculating a triangle's area in space, the cross product helps find a vector that is perpendicular to the plane defined by two vectors.
The formula to calculate the cross product of two vectors \( \overrightarrow{A} = (a_1, a_2, a_3) \) and \( \overrightarrow{B} = (b_1, b_2, b_3) \) is given by:
\[ \overrightarrow{A} \times \overrightarrow{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} = (a_2b_3 - a_3b_2)\mathbf{i} - (a_1b_3 - a_3b_1)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k} \]
This determinant style calculation provides us with a new vector whose direction is perpendicular to both original vectors, aligning with the right-hand rule of vector cross products. The length or magnitude of this vector is proportional to the sine of the angle between the original vectors, providing insightful information about the relationship and relative positioning of the vectors.

Vektormagnitude

Understanding "Vektormagnitude," or vector magnitude, is essential in many vector operations, particularly when dealing with physical properties such as length or displacement. A vector's magnitude is a scalar value that represents the length of the vector, irrespective of its direction. This is crucial when calculating properties like the area of a triangle.
The formula for the magnitude of a vector \( \overrightarrow{v} = (v_1, v_2, v_3) \) is:
\[ ||\overrightarrow{v}|| = \sqrt{v_1^2 + v_2^2 + v_3^2} \]
Here, each component of the vector is squared, summed, and square-rooted to give the overall length of the vector. When we calculate the magnitude of a cross product vector, it reflects how much the two original vectors "encompass" mutually in space, providing the necessary scalar for calculations like the area of a triangle. In practical terms, knowing a vector's magnitude aids in assessing and interpreting spatial relationships within vector fields.