Problem 1 Write the equation of the line p... [FREE SOLUTION]

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Linear Algebra: A Modern Introduction

David Poole

$Math Studyset 91影视 Explanations$ Math

4 Edition

Chapter 1: Problem 1

Write the equation of the line passing through P with normal vector n in (a) normal form and (b) general form. $P=(0,0), \mathbf{n}=\left[\begin{array}{l}3 \\ 2\end{array}\right]$

Short Answer

Expert verified

The line equation is $3x + 2y = 0$ in both normal and general forms.

Step by step solution

Understanding the Problem

You need to find the equation of a line passing through a point $ P = (0,0) $ with a given normal vector $ \mathbf{n} = \begin{bmatrix} 3 \ 2 \end{bmatrix} $. This equation should be derived both in the normal form and general form.

Using Normal Form of Line Equation

The normal form of a line equation is given as $ ax + by = c $, where $(a, b)$ is the normal vector. Substitute $ a = 3 $ and $ b = 2 $. Since the line passes through $(0, 0)$, the equation becomes $ 3 \times 0 + 2 \times 0 = c $. Thus, $ c = 0 $. The normal form of the line is $ 3x + 2y = 0 $.

Converting to General Form

The general form of a line is $ Ax + By + C = 0 $. Notes from the normal form indicate $ A = 3 $, $ B = 2 $, and $ C = 0 $. Therefore, the general form of the line equation is simply $ 3x + 2y + 0 = 0 $, which can also be written as $ 3x + 2y = 0 $.

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Key Concepts

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

Normal Form

The normal form of a line is a way to express the equation of a line using a normal vector. This vector is perpendicular to any vector lying on the line. The equation in normal form is depicted as $ ax + by = c $. In this formulation:

$ (a, b) $ represents the components of the normal vector $ \mathbf{n} $.
$ c $ is a constant that usually represents the distance from the origin to the line, if the line doesn't pass through the origin.

Given that the line passes through the point $(0, 0)$, it simplifies the equation since both $ x $ and $ y $ are zero, making $ c = 0 $. Hence, for a line through the origin with a normal vector $ \mathbf{n} = \begin{bmatrix} 3 \ 2 \end{bmatrix} $, the normal form becomes $ 3x + 2y = 0 $. This tells us that every point on this line yields the same equation when substituting the $ x $ and $ y $ terms, validating the line's form.

General Form

The general form of a line is an expression that's often preferred because of its adaptability in various calculations. The equation is expressed as $ Ax + By + C = 0 $. Here:

$ A $ and $ B $ correspond to the components of the normal vector of the line.
$ C $ represents the constant term that shifts or places the line within the coordinate plane.

In this scenario, because our line goes through the origin $(0, 0)$, $ C $ consequently remains $ 0 $, simplifying our equation to $ 3x + 2y = 0 $. The significance of the general form is that it shows any linear relation between $ x $ and $ y $, allowing for a versatile description that can be manipulated algebraically for intersection or parallel assessment.

Normal Vector

A normal vector is a key component when working with line equations in both the normal and general forms. This vector $ \mathbf{n} $ is a perpendicular vector to the line it's associated with. Its components $ (a, b) $ play a vital role in determining the direction and slope of the line:

The vector $ \mathbf{n} $ defines that any vector along the line is orthogonal to it, meaning their dot product is zero.
In our current example, $ \mathbf{n} = \begin{bmatrix} 3 \ 2 \end{bmatrix} $, which helps rapidly establish the line equation parameters.

By understanding the concept of normal vectors, one can easily derive the line鈥檚 equation by inserting its components $ (a, b) $ directly into both the normal form and general form of the equation.

Line Through a Point

A line defined through a specific point forms the basis for drafting its equation in normal and general forms. In our exercise, the line passes through the origin point $ P = (0, 0) $, a special and often simplifying scenario.

When a line passes through the origin, calculations become simpler because both the $ x $ and $ y $ values are zero.
This condition directly influences the constant term of our equations, transitioning it to zero, which simplifies both the normal and general forms to omit a separate constant.

Comprehending how a point affects line equations is crucial because it helps in identifying particular solutions and simplifying general expressions, making your calculations less cumbersome.

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91影视

Write the equation of the line passing through P with normal vector n in (a) normal form and (b) general form. \(P=(0,0), \mathbf{n}=\left[\begin{array}{l}3 \\ 2\end{array}\right]\)

Short Answer

Step by step solution

Understanding the Problem

Using Normal Form of Line Equation

Converting to General Form

Key Concepts

Normal Form

General Form

Normal Vector

Line Through a Point

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