**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...

See similar textbooks

Related questions

Q: What is the horizontal of line through the point (4,-6)?

A: Solution: In the XY coordinate system, the point notation is always written as (x,y) And, x…

Q: 9. What are the coordinates of A' after AB is translated by the rule (x, y) → (x + 5, y – 2)? A 4- 3…

A: As per our guidelines I am solving only first problem please repost other question. 9. A' after AB…

Q: What are the coordinates of P(23t/2)? А. (-1, 0) В. (1, 0) С. (0,1) D.(0,-1)

A: Solution

Q: ). makes equal intercepts on the axes and passes through the point (2,3). 5. passes through the…

A: (a) Let the equation of the line bexa+yb=1Since it makes equal intercepts on the coordinates axes,we…

Q: 2- A line through point (-3,4) is perpendicular to the line y = 2x - 3. Find the point where the…

A: Point of intersection is (11/5, 7/5)

Q: Find x such that the line through (2, - 3) and (3, 2) is perpendicular to the line through (- 2, 4)…

A: We have the points A(2,-3),B(3,2),C(-2,4) and D(x,-1) We have to find the value of x such that the…

Q: Find a set of symmetric equations for the line “A" perpendicular to the lines having direction…

Q: Compute the distance from the point (3, 0, -1) to the line containing the points (1,0,−4) and (1,…

Q: Which 2 points are on the line (x,y) = (3,7) + k(-1,2), k E R Select one: (-1, 2), (3, 7) (3,7), (2,…

Q: 2. Sketch the line = (-4,-4) + t(6,8).

A: To sketch the line r=(-4,-4)+t(6,8)

Q: 25. (17,-7) (-8, 11) Parallel: Perpendicular:

Q: The coordinates of the missing point are A. (4, 4) В. (5, 4) (-1,4) (?, ?) С. (6, 4) D. (7, 4) x+ 3…

A: topic- sketch of a figure distance between x=3 and (-1,4) is 3-(-1)=3+1=4 x-coordinates of missing…

Q: True or False: (a) The lines y = 2x - 3 and y = -2x - 3 are perpendicular (b) The lines y = 2/5x +…

Q: What is the equation of the line which passes through (2,1) and (-1,2)? a) x+3y+5=0 b) -x+3y-5=0…

Q: Find the line perpendicular to y=2x-3 and through the point (2,-3) B) find the lines both parallel…

Q: Find the distance between each pair of points: с) (-4, 1) and (4, 5) d) (x — 2, у —3) and (х + 4, у…

A: Distance between two points Ax1, y1 and Bx2, y2 is, AB=x2-x12+y2-y12

Q: 29. (-2, 2) reflected through the y-axis is a. (-2, 2) b. (2, 2) с. (-2, -2) d. (2, -2)

A: Rule:- A point ( x,y ) After reflection through y axis becomes ( -x , y)

Q: Assume a line y = mx + c, [asses through the points (1, 15) and (-3, -5) Determine the slope, m, of…

Q: 10) Write the equation of the line through (8, 45) that is perpendicular to y = 7.

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} \

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step by step

Solved in 4 steps with 4 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Cartesian Coordinates$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Similar questions

Find the equation of the straight line that (1). makes equal intercepts on the axes and passes through the point (2, 3). (2). passes through the point (-5, 4) and is such that the portion intercepted between the axes is divided by the point in the ratio 1:2.
The point on the line y= 2x+3 that is closest to the point (4,2) is (a,b). Then 5b – 20a= O a. 2 O b. 3 О с. 5 O d. 15 O e. 11 Of. 1 О д. 13 O h. 7 O i. 9
At what point on the line y = -b does the line segment from (0, 0) to (a,0) subtend the greatest angle? (0,5) レゅーb
Find the equation of the straight line that(1).makes equal intercepts on the axes and passes through the point (2,3).(2).passes through the point (-5,4) and is such that the portion intercepted between the axes is divided by thepoint in the ratio 1:2.
Find the equation of the straight line that(a).makes equal intercepts on the axes and passes through the point (2,3).(b).passes through the point (-5,4) and is such that the portion intercepted between the axes is divided by thepoint in the ratio 1:2.