**Title: Finding the Point of Intersection of Two Lines in R²****Problem Statement:**The goal is to find the point of intersection of the lines defined by the parametric equations:\[ \mathbf{r_1}(t) = \langle -1, 1 \rangle + t \langle 8, 4 \rangle \]and\[ \mathbf{r_2}(s) = \langle 2, 1 \rangle + s \langle 8, 6 \rangle \]in $\mathbb{R}^2$.**Expression for Point of Intersection:**\[ (x, y) = \left( \text{input box} \right) \]**Explanation:**We need to determine the values of $t$ and $s$ for which the two parametric equations yield the same coordinates $(x, y)$.1. **Equation (1) from $\mathbf{r_1}(t)$:** \[ x_1(t) = -1 + 8t \] \[ y_1(t) = 1 + 4t \]2. **Equation (2) from $\mathbf{r_2}(s)$:** \[ x_2(s) = 2 + 8s \] \[ y_2(s) = 1 + 6s \]To find the intersection point, set $x_1(t) = x_2(s)$ and $y_1(t) = y_2(s)$:\[-1 + 8t = 2 + 8s \tag{1}\]\[1 + 4t = 1 + 6s \tag{2}\]Solving these equations will yield the values of $t$ and $s$ for which the lines intersect. The corresponding coordinates can then be obtained by substituting $t$ back into $\mathbf{r_1}(t)$ or $\mathbf{r_2}(s)$.**Submission Box:**Here, the coordinates of the intersection point can be entered:\[ (x, y) = \left( \begin{matrix}\ \\\ \\\end{matrix} \

Consider the given lines in polar form. To find the intersection points equate the x and y…

Answered: Find the point of intersection of the…

Math

Calculus

Find the point of intersection of the lines r;(t) = (-1, 1) + t(8, 4) and rɔ(s) = (2, 1) + s(8, 6) in R?. (х, у) -

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Concept explainers

Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.

Question

**Title: Finding the Point of Intersection of Two Lines in R²**

**Problem Statement:**
The goal is to find the point of intersection of the lines defined by the parametric equations:
\[ \mathbf{r_1}(t) = \langle -1, 1 \rangle + t \langle 8, 4 \rangle \]
and
\[ \mathbf{r_2}(s) = \langle 2, 1 \rangle + s \langle 8, 6 \rangle \]
in $\mathbb{R}^2$.

**Expression for Point of Intersection:**
\[ (x, y) = \left( \text{input box} \right) \]

**Explanation:**
We need to determine the values of $t$ and $s$ for which the two parametric equations yield the same coordinates $(x, y)$.

1. **Equation (1) from $\mathbf{r_1}(t)$:**
\[
x_1(t) = -1 + 8t
\]
\[
y_1(t) = 1 + 4t
\]

2. **Equation (2) from $\mathbf{r_2}(s)$:**
\[
x_2(s) = 2 + 8s
\]
\[
y_2(s) = 1 + 6s
\]

To find the intersection point, set $x_1(t) = x_2(s)$ and $y_1(t) = y_2(s)$:
\[
-1 + 8t = 2 + 8s \tag{1}
\]
\[
1 + 4t = 1 + 6s \tag{2}
\]

Solving these equations will yield the values of $t$ and $s$ for which the lines intersect. The corresponding coordinates can then be obtained by substituting $t$ back into $\mathbf{r_1}(t)$ or $\mathbf{r_2}(s)$.

**Submission Box:**
Here, the coordinates of the intersection point can be entered:
\[ (x, y) = \left( \begin{matrix}
\ \\
\ \\
\end{matrix} \

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