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Chapter 8: Problem 36

Testing for Collinear Points In Exercises \(35-40\) , use a determinant to determine whether the points are collinear. $$(3,-5),(6,1),(4,2)$$

Short Answer

Expert verified
The points (3,-5), (6,1), and (4,2) are not collinear.

Step by step solution

01

Set Up the Determinant

First, set up the determinant with the coordinates of the points as follows: \[\begin{vmatrix}3 & -5 & 1 \6 & 1 & 1 \4 & 2 & 1 \end{vmatrix}\]
02

Calculate the Determinant

Next, calculate the determinant. The formula for 3 x 3 determinant is given by: \[\begin{vmatrix}a & b & c \d & e & f \g & h & i \end{vmatrix}= aei + bfg + cdh - ceg - bdi - afh\] So applying this formula, we get: (3*1*1 + (-5)*1*1 + 1*6*2) - (1*1*4 + (-5)*6*1 + 3*1*2) = -4
03

Interpret the Result

The determinant value is -4, which is not equal to zero. Therefore, the three points are not collinear.

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Key Concepts

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

Determinant
A determinant provides a unique value computed from a square matrix and can help solve systems of linear equations, find area, and more. It plays a crucial role in determining whether points in geometry are collinear. If you calculate the determinant for a set of points and the result is zero, the points are collinear, meaning they lie on the same straight line. However, if the determinant is not zero, the points are not collinear. This mathematical tool simplifies complex calculations involving geometric planes and helps in bringing solutions to problems that involve several variables at once.
3x3 Matrix
A 3x3 matrix is a representation of a set of numbers in a grid format with three rows and three columns. For determining the collinearity of points in a 3D space, this matrix comes in handy. With three points having coordinates \(x, y\) in a 2D plane, you can set a 3x3 matrix like this:
  • The three pairs of x and y coordinates fill the first two columns.
  • An extra column of ones makes up the matrix.
With this setup, the calculation involves evaluating the determinant of the 3x3 matrix to check if it equals zero, which implies collinearity. This method simplifies understanding geometric relationships and aids in various spatial calculations.
Precalculus
Precalculus serves as the bridge to understanding more advanced mathematical concepts. It introduces students to a variety of functions and their properties, as well as fundamental geometry and trigonometry skills. In the realm of precalculus, determining collinearity using determinants represents an integration of algebra and geometry concepts, offering a practical use case for these ideas. Here, understanding how to compute and interpret a determinant is crucial, as it offers insights into relationships between points in space, setting a foundation for more complex topics in calculus and beyond.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, studies geometric figures through a system of coordinates. This approach allows mathematicians and students alike to analyze and visualize the properties of shapes by using algebraic techniques:
  • It enables the analysis of lines, curves, and their properties using equations.
  • Collinearity tests like finding a determinant help understand positional relationships among points.
In coordinate geometry, the collinearity of points signifies that points lie on the same straight line, which is often determined through the calculation of a determinant. This gives a clear and computable method to ascertain geometric relationships.

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Most popular questions from this chapter

In Exercises \(89-94\) evaluate the determinant in which the entries are functions. Determinants of this type occur when change of variables are made in calculus. $$\left| \begin{array}{ll}{x} & {\ln x} \\ {1} & {1 / x}\end{array}\right|$$

Properties of Determinants In Exercises \(101-103\) ,a property of determinants is given \((A\) and \(B\) are squarematrices). State how the property has been applied to the given determinants and use a graphing utility to verify the results. If \(B\) is obtained from \(A\) by multiplying a row by a nonzero constant \(c\) or by multiplying a column by a nonzero constant \(c,\) then \(|B|=c|A|\) $$\left.\begin{array}{r|rr}{\text { (a) }} & {5} & {10} \\ & {2} & {-3}\end{array}\right|=5 \left| \begin{array}{rr}{1} & {2} \\ {2} & {-3}\end{array}\right|$$ $$(b)\left|\begin{array}{rrrr}{1} & {8} & {-3} \\ {3} & {-12} & {6} \\ {7} & {4} & {9}\end{array}\right|=12 \left| \begin{array}{rrr}{1} & {2} & {-1} \\\ {3} & {-3} & {2} \\ {7} & {1} & {3}\end{array}\right|$$

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination. $$\left\\{\begin{aligned} x &-3 z=-2 \\ 3 x+y-2 z &=5 \\ 2 x+2 y+z &=4 \end{aligned}\right.$$

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system. $$\left\\{\begin{aligned} x+2 y+2 z+4 w=& 11 \\ 3 x+6 y+5 z+12 w=& 30 \\ x+3 y-3 z+2 w=&-5 \\ 6 x-y-z+\quad w=&-9 \end{aligned}\right.$$

A corporation has three factories, each of which manufactures acoustic guitars and electric guitars. The production levels are represented by \(A\) . \(A=\left[ \begin{array}{ccc}{70} & {50} & {25} \\ {35} & {100} & {70}\end{array}\right]\) (a) Interpret the value of \(a_{22}\) (b) How could you find the production levels when production is increased by 20\(\% ?\) (c) Each acoustic guitar sells for \(\$ 80\) and each electric guitar sells for \(\$ 120 .\) How could you use matrices to find the total sales value of the guitars produced at each factory?
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