Problem 51 - Algebra and Trigonometry

Question

Determinants are used to show that three points lie on the same line (are collinear). If $$ \left|\begin{array}{lll} {x_{1}} & {y_{1}} & {1} \\ {x_{2}} & {y_{2}} & {1} \\ {x_{3}} & {y_{3}} & {1} \end{array}\right|=0 $$ then the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) are collinear. If the determinant does not equal \(0,\) then the points are not collinear. Use this information to work Exercises \(51-52\) Are the points \((3,-1),(0,-3),\) and \((12,5)\) collinear?

Step-by-Step Solution

Answer

Yes, the points (3,-1), (0,-3), and (12,5) are collinear.

Step 1: Create the Matrix

First, it's necessary to arrange the given pair of points in a 3x3 matrix: \[ \begin{array}{ccc} 3 & -1 & 1 \ 0 & -3 & 1 \ 12 & 5 & 1 \end{array} \]

Step 2: Calculate the Determinant

The next step is to calculate the determinant of this matrix. Using the formula for the determinant of a 3x3 matrix, the result will be: \( Determinant = 3[(-3*1) - (5*1)] - (-1)[(0*1) - (12*1)] + 1[(0*5) - (-3*12)] = -24 - 12 + 36 = 0 \)

Step 3: Interpret the Result

Since the determinant equals to 0, it indicates that the given points are collinear.

Key Concepts

Determinants in Algebra

Determinants play a crucial role in algebra, especially when it comes to analyzing the relationships between points in a coordinate system. A determinant is a special number that can be calculated from the elements of a square matrix. The value of a determinant provides important information about the matrix it was derived from. For example, it can indicate whether a system of linear equations has a unique solution, whether a matrix is invertible, and as the exercise mentions, whether points in a plane are collinear.

When dealing with points and their collinearity, determinants offer a simple yet effective method for verification. In the context of linear algebra, a determinant equaling zero signifies that the vectors (or points in a plane) being examined are linearly dependent, which, in turn, indicates that they lie on the same line, thereby showing collinearity.

Matrix for Collinearity

As highlighted in the exercise, collinearity of points can be determined using a matrix and the concept of determinants. To check if three points in a two-dimensional space are collinear, one can use a 3x3 matrix where each row represents a point with its x and y coordinates supplemented by a 1 in the third column. This 1 represents the homogenous coordinate, which in projective geometry helps handle points at infinity but in the case of collinearity ensures the matrix is square.

The use of this specific 3x3 matrix is significant. Its determinant reflects the area of the triangle formed by the points. If the points are collinear, the area of this 'triangle' is zero - hence the determinant is zero. This property provides a direct and algebraic route to ascertain whether given points lie on the same straight line or not without the need for graphing or additional geometric reasoning.

3x3 Matrix Determinant Calculation

Calculating the determinant of a 3x3 matrix is a fundamental skill in algebra that involves following a set procedure. This skill is applied in the exercise for determining collinearity. For a general 3x3 matrix, the determinant can be calculated using a method known as the Rule of Sarrus or the cofactor expansion.

Rule of Sarrus:

In the Rule of Sarrus, you only need to remember a pattern of multiplication across the diagonals of the matrix, and then subtract the product of diagonals going in the opposite direction. While simple for hand calculations, this method has limited use and is not applicable to larger matrices.

Cofactor Expansion:

The cofactor expansion is more widely used and involves calculating the determinant by expanding along a row or column and taking the sum of the products of each element and its corresponding minor's determinant, often indicated by a sign change based on position.

In the textbook solution, a determinant calculation similar to the cofactor expansion has been used, effectively showing the process of validating the collinearity of points mathematically.