/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 Find the equation of a line thro... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 5

Find the equation of a line through (2,-3) that is parallel to the line with equation \(r=3 i-7 j+\lambda(4 i-3 j)\).

Short Answer

Expert verified
The equation of the line is \(y = -\frac{3}{4}x - \frac{3}{2}\).

Step by step solution

01

Understand the Given Line

The given line is expressed in vector form, with equation \( \mathbf{r} = 3\mathbf{i} - 7\mathbf{j} + \lambda(4\mathbf{i} - 3\mathbf{j}) \). This represents a line through the point \((3, -7)\) and in the direction of \((4, -3)\).
02

Identify the Slope of the Given Line

The direction vector of the line is \((4, -3)\). The slope of a line having the direction vector \((a, b)\) is \( \frac{b}{a} \). Therefore, the slope of this line is \(-\frac{3}{4}\).
03

Use the Slope for the Parallel Line

Lines that are parallel have the same slope. Hence, the slope of the line through \((2, -3)\) that is parallel to the given line is also \(-\frac{3}{4}\).
04

Apply the Point-Slope Form

The point-slope form of a line's equation is \(y - y_1 = m(x - x_1)\). Here, \((x_1, y_1) = (2, -3)\) and the slope \(m = -\frac{3}{4}\). Substitute these values to get: \[y + 3 = -\frac{3}{4}(x - 2)\].
05

Simplify to Find the Line's Equation

Expand and simplify the equation: \[y + 3 = -\frac{3}{4}x + \frac{3}{2}\]. Subtract 3 from both sides to get: \[y = -\frac{3}{4}x + \frac{3}{2} - 3\]. Further simplification gives: \[y = -\frac{3}{4}x - \frac{3}{2}\].

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Equations
Vector equations are a way to represent lines in a more geometric form. A vector equation of a line typically includes a starting point vector and a direction vector. For instance, in the problem at hand, the line is given as \( \mathbf{r} = 3\mathbf{i} - 7\mathbf{j} + \lambda(4\mathbf{i} - 3\mathbf{j}) \). This is interpreted as:
  • \( 3\mathbf{i} - 7\mathbf{j} \), the point from which the line passes.
  • \( \lambda(4\mathbf{i} - 3\mathbf{j}) \), a direction vector scaled by a parameter \( \lambda \).
The direction vector tells us the line's trajectory in relation to the coordinate axes. When asked for a parallel line, this direction vector plays a crucial role since the parallel line will have the same direction vector. Vector equations are instrumental due to their compact representation and ease of use in calculations.
Slope-Intercept Form
The slope-intercept form is a popular way to express the equation of a line. It is represented as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form is straightforward because it openly tells us:
  • How steep the line is through its slope \( m \).
  • Where the line crosses the y-axis, indicated by \( b \).
For our given problem, converting the simplified equation \( y = -\frac{3}{4}x - \frac{3}{2} \) into the slope-intercept form shows that the slope \( m \) is \( -\frac{3}{4} \) and the y-intercept \( b \) is \( -\frac{3}{2} \).
Knowing the slope makes understanding the line's behavior intuitive. Negative slopes imply that as \( x \) increases, \( y \) decreases, which visually tilts the line downwards from left to right.
Point-Slope Form
The point-slope form is particularly effective for writing equations of lines when you're given a point and a slope. It is expressed as \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1)\) is a known point on the line, and \( m \) is the slope. This form is advantageous because:
  • It's easy to construct using a specific point and slope.
  • It helps transition to other forms, like the slope-intercept form, by simplification.
For the exercise, using the point \((2, -3)\) and slope \( -\frac{3}{4} \), the initial equation is set up. By applying the point-slope form, we get:\[ y + 3 = -\frac{3}{4}(x - 2) \]This equation can then be transformed into either vector or slope-intercept forms by further simplification, enhancing its versatility for solving and analyzing the problem.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find \(m\) such that the following vectors are coplanar; otherwise, show that it is not possible. $$\mathbf{u}=(2,-3,2 m), \mathbf{v}=(m,-3,1), \mathbf{w}=(1,3,-2)$$

Show that \(|\mathbf{u} \times \mathbf{v}|=\sqrt{|\mathbf{u}|^{2}|\mathbf{v}|^{2}-(\mathbf{u} \cdot \mathbf{v})^{2}}\).

Find the volume of the parallelepiped with \(\mathbf{u}, \mathbf{v}\) and \(\mathbf{w}\) as adjacent edges. $$\mathbf{u}=4 \mathbf{i}+2 \mathbf{j}+3 \mathbf{k}, \mathbf{v}=5 \mathbf{i}+6 \mathbf{j}+2 \mathbf{k}, \mathbf{w}=2 \mathbf{i}+3 \mathbf{j}+5 \mathbf{k}$$

For what value of \(t\) is the vector \(2 t i+4 j-(10+t)\) k perpendicular to the vector \(1+i j+k ?\)

What can you conclude about the angle between two non-zero vectors \(\mathbf{u}\) and \(\mathbf{v}\) if \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u} \times \mathbf{v}| ?\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.