Math Problem Statement
Können sie folgende Aufgabe für mich lösen?
Solution
The exercise asks you to fill in the missing values so that the point lies on the line .
Part a)
The line is described by the parametric equation:
\begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix} + r \cdot \begin{pmatrix} 1 \\ 3 \\ -1 \end{pmatrix}$$ The point $$R$$ is $$R(-1, -3.5, \square)$$, where we need to find the missing coordinate. To find $$r$$, we equate the first two coordinates of $$R$$ to the corresponding coordinates in the parametric equation of the line. 1. For the first coordinate: $$2 + r \cdot 1 = -1$$ Solving for $$r$$: $$r = \frac{-1 - 2}{1} = -3$$ 2. For the second coordinate: $$1 + r \cdot 3 = -3.5$$ Substitute $$r = -3$$: $$1 + (-3) \cdot 3 = -3.5$$ $$1 - 9 = -8$$ This is not equal to $$-3.5$$, so there seems to be an inconsistency here. Are you sure about the value of $$-3.5$$? Could this be a misprint? Assuming the task is consistent, we can use the first equation to solve for the missing $$z$$-coordinate: $$0 + r \cdot (-1) = z$$ Substitute $$r = -3$$: $$0 + (-3) \cdot (-1) = z$$ $$z = 3$$ Thus, $$R$$ is $$(-1, -3.5, 3)$$, assuming $$-3.5$$ is correct. --- ### Part b) The line is given by: $$g: \vec{x} = \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix} + r \cdot \begin{pmatrix} 1 \\ \text{?} \\ 2 \end{pmatrix}$$ The point $$R$$ is $$R(4, 5, \text{?})$$. 1. For the first coordinate: $$3 + r \cdot 1 = 4$$ Solving for $$r$$: $$r = \frac{4 - 3}{1} = 1$$ 2. For the second coordinate, we need to find the missing value in the direction vector. Let that value be $$b$$. The second coordinate gives: $$1 + r \cdot b = 5$$ Substitute $$r = 1$$: $$1 + b = 5$$ $$b = 4$$ 3. For the third coordinate: $$-1 + r \cdot 2 = z$$ Substitute $$r = 1$$: $$-1 + 2 = z$$ $$z = 1$$ Thus, the direction vector is: $$\begin{pmatrix} 1 \\ 4 \\ 2 \end{pmatrix}$$ And the point $$R$$ is $$(4, 5, 1)$$. --- Do you need more details or clarification on any step? ### Related Questions: 1. How do you verify if a point lies on a line in parametric form? 2. Can you extend the parametric form to describe planes or higher-dimensional objects? 3. How would you calculate the distance between a point and a line? 4. What if a line is given in symmetric form? How do you solve it then? 5. How can you handle inconsistent systems when solving parametric equations? ### Tip: Always double-check the consistency of given coordinates, especially when working with parametric equations, as errors can propagate through the solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Vector Algebra
Parametric Equations
Lines in 3D
Formulas
Parametric equation of a line: \( \vec{x} = \vec{a} + r \cdot \vec{b} \)
Theorems
Condition for a point to lie on a parametric line: Solve for the parameter \(r\) using the coordinates.
Suitable Grade Level
Grades 11-12
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