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Chapter 9: Problem 37

Find the equation of the line through \((1,-2)\) and perpendicular to \([4,1]^{\prime}\).

Short Answer

Expert verified
The equation of the perpendicular line is \(y = -4x + 2\).

Step by step solution

01

Understand the Slope of a Vector

The vector \([4,1]'\) represents the direction of a line. The slope \(m_1\) of this line can be calculated from this vector as \(m_1 = \frac{y}{x} = \frac{1}{4}\). This is the slope of the line parallel to the given vector.
02

Identify the Slope of the Perpendicular Line

The slope of a line perpendicular to another line with slope \(m_1\) is the negative reciprocal of \(m_1\). Therefore, the slope \(m_2\) of the line perpendicular to the given line \([4,1]'\) is \(m_2 = -\frac{1}{m_1} = -4\).
03

Use Point-Slope Form to Find Equation

To find the equation of a line with slope \(-4\) passing through the point \((1,-2)\), use the point-slope form of a line equation: \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (1,-2)\) and slope \(m = -4\).
04

Plugpoint into Point-Slope Formula

Substitute the values into the point-slope formula: \(y - (-2) = -4(x - 1)\). Simplify this to \(y + 2 = -4x + 4\).
05

Rearrange into Slope-Intercept Form

To express the equation in slope-intercept form \(y = mx + b\), solve for \(y\):\[y = -4x + 4 - 2\] which simplifies to \[y = -4x + 2\].

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Key Concepts

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

Slope of Vector
Vectors are like arrows. They have both a direction and a magnitude. When we talk about the slope of a vector in a 2D space, we are primarily concerned with its direction. The vector \([4, 1]\) tells us how far to move along the x-axis and y-axis.

To find the slope of a line that follows this direction, we use the formula for slope:
  • Multiply the vector components: \(m = \frac{\text{change in } y}{\text{change in } x}\)
  • In our example, the vector is \([4, 1]\). So break it down: the change in y is 1, and change in x is 4.
  • Using the formula, the slope \(m = \frac{1}{4}\).
A vector's slope gives you an intuitive idea of how steep (or flat) the line heading this direction is.
Negative Reciprocal Slope
When two lines are perpendicular, their slopes are connected in a special way. They are negative reciprocals of each other.

What does that mean? If you have a slope \(m_1\), the perpendicular slope \(m_2\) can be found using these steps:
  • First, find the reciprocal: if \(m_1 = \frac{1}{4}\), the reciprocal is \(\frac{4}{1}\) or simply 4.
  • Next, make it negative: the negative reciprocal of 4 is \(-4\).
  • This means our perpendicular slope \(m_2 = -4\).
Understanding negative reciprocal slopes is crucial. It helps in determining how lines intersect at exactly ninety degrees, like the corners of a square.
Point-Slope Form
The point-slope form is like a blueprint for building a line. It's a way to write the equation when you have one point the line passes through and the slope of the line.Here is the point-slope formula:
  • Simplified, it looks like this: \(y - y_1 = m(x - x_1)\).
  • In our scenario, you have a point \((1, -2)\) and a slope \(-4\).
  • Plug these values into the formula to get \(y - (-2) = -4(x - 1)\).
Simplifying this will give you a clean line equation:
  • Distribute \(-4\) over \((x - 1)\): \(y + 2 = -4x + 4\).
  • Move constant terms to solve for \(y\): \(y = -4x + 2\).
This form is practical. It quickly translates known information into a usable format for graphing or further calculation.

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Most popular questions from this chapter

Where do a plane through \((2,0,-1)\) and perpendicular to \(\left[\begin{array}{r}-1 \\ 1 \\ 3\end{array}\right]\) and a line through the points \((1,0,-2)\) and \((1,-1,1)\) intersect?

Find the length of \(\mathbf{x}=[-2,7]^{\prime}\).

Assume that a population is divided into three age classes and that \(80 \%\) of the females age 0 and \(10 \%\) of the females age present at time \(t\) survive until time \(t+1\). Assume further that females age 1 have an average of \(1.6\) female offspring and females age 2 have an average of \(3.9\) female offspring. If, at time 0 , the population consists of 1000 females age 0,100 females age 1 , and 20 females age 2 , find the Leslie matrix and the number of females in each age class at time \(3 .\)

Let \(A=(-1,0)\) and \(B=(2,-3)\). Find the vector representation of \(\overrightarrow{A B}\).

Suppose that $$ L=\left[\begin{array}{ll} 3 & 2 \\ 1.5 & 1 \end{array}\right] $$ is the Leslie matrix for a population with two age classes. (a) Determine both eigenvalues. (b) Give a biological interpretation of the larger eigenvalue. (c) Find the stable age distribution.
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