Math Problem Statement
Solution
I will analyze and solve the tasks provided in your document. Here's a detailed breakdown for each task:
Aufgabe 1: Linearkombination für den jeweils ersten Vektor
a) Matrix:
0.6 & -9 \\ 1.4 & -21 \\ -1 & -15 \end{bmatrix}$$ Check if $$[0.6, 1.4, -1]$$ is a linear combination of $$[-9, -21, -15]$$. This means solving the equation: \[ c_1 \cdot [-9, -21, -15] = [0.6, 1.4, -1] \] #### b) Matrix: $$\begin{bmatrix} 0.4 & 0.6 & -0.4 \\ 0.6 & 0.8 & 2 \\ 0.2 & 0.2 & -0.7 \end{bmatrix}$$ Check if $$[0.4, 0.6, 0.2]$$ is a linear combination of $$[0.6, 0.8, 0.2]$$ and $$[-0.4, 2, -0.7]$$. #### c) Vectors: $$\begin{bmatrix} 5 & -2 & 3 \\ 1.25 & 5 & -21.25 \\ 2.5 & 3 & -14.5 \end{bmatrix}$$ Check if $$[5, 1.25, 2.5]$$ is a linear combination of the other two vectors. --- ### Aufgabe 2: Drohnenflug #### a) Gerade durch Punkte $$A(100, -50, 0)$$ und $$B(200, 20, 15)$$: Find the equation of the line using: $$\vec{x} = \vec{U} + r \cdot \vec{V}$$ where: $$\vec{U} = (100, -50, 0), \quad \vec{V} = (200 - 100, 20 - (-50), 15 - 0) = (100, 70, 15)$$ #### b) Punkte $$C$$ und $$D$$: Substitute $$r = 2$$ and $$r = 3$$ into the line equation to find the coordinates of $$C$$ and $$D$$. #### c) Flugzeugbewegung: The starting point is $$P(-200, 800, 45)$$ and the direction vector is $$\vec{W} = (300, -320, 0)$$. The equation is: $$\vec{x} = \vec{P} + t \cdot \vec{W}$$ Find the point $$Q$$ at $$t = 2$$. #### d) Punkte $$D$$ und $$Q$$: Verify if the coordinates of $$D$$ and $$Q$$ are the same and interpret the relationship between their motions. #### e) Fluggeschwindigkeit: Express the movement in terms of the plane $$x_1 - x_2 =$$ constant and calculate the speed in $$m/s$$. --- Would you like me to solve these step-by-step or focus on specific sub-tasks?Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Vector Arithmetic
Equation of a Line
Parametric Equations
Relative Motion
Formulas
Line equation: x = U + r * V
Linear combination: c1 * V1 + c2 * V2 = Target Vector
Speed formula: v = distance/time
Theorems
Linear Independence of Vectors
Vector Addition and Scalar Multiplication
Distance Formula
Suitable Grade Level
Grades 10-12
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