Question

Use a determinant to determine whether the points are collinear. $$ (-3,-4),(-1,-1),(5,5) $$

Short answer

The points (-3,-4), (-1,-1), and (5,5) are collinear.

Step-by-step solution

Set up the matrix

For given points \((-3,-4), (-1,-1), (5,5)\), create a 3x3 matrix by inputting the given points coordinates on its row for the first and second columns. Add \(1\) as the value for each of the entries in the third column. To visualize, the matrix becomes: \[ \left[ \begin{array}{ccc} -3 & -4 & 1 \ -1 & -1 & 1 \ 5 & 5 & 1 \end{array}\right] \]

Calculate the determinant

Calculate the determinant of the matrix. The determinant of the matrix is found using the formula \[ \text{det}(A) = aei + bfg + cdh - ceg - bdi - afh \]. Applying this to the matrix, the determinant comes out as: \[ -3 * -1 * 1 + -4 * 1 * 5 + 1 * -1 * 5 - 1 * -1 * 1 - -3 * 1 * 5 - -4 * -1 * 1 \]

Interpret the result

Solving the previously calculated expression results in \(0\). If the determinant is \(0\), this means the points are collinear, meaning they all lie on the same line. If the determinant isn't \(0\), it signifies the points are not collinear.

Key concepts

Matrix Determinant

The determinant of a matrix is a special number that can be calculated from its elements. In a 2x2 matrix, the determinant is calculated as \( ad - bc \). However, for a 3x3 matrix, which often comes into play for higher-dimensional problems, the calculation is more complex.

The determinant of a 3x3 matrix A, often denoted as \( det(A) \), can be calculated using the formula \( aei + bfg + cdh - ceg - bdi - afh \) where \( a, b, c, \) etc., are the elements of the matrix. Visually, this entails taking the sum of products of the diagonals from the left top to the right bottom and subtracting the sum of products of the diagonals from the right top to the left bottom.

Significance in Collinearity

In the context of collinearity, the determinant serves as a decisive tool. If the determinant of the matrix formed by the coordinates of the points (each adding 1 as the third coordinate) is zero, this implies that the area of the triangle formed by the points is zero. Consequently, the points must be collinear—lying on a single straight line. If the determinant is not zero, the points form a triangle with a non-zero area, indicating that they are not collinear.

Collinearity in Algebra

Collinearity in algebra refers to a condition where three or more points lie on a single straight line. This concept is vital in coordinate geometry and has implications in various algebraic problems and analyses.

Determining collinearity algebraically involves various methods, with matrix determinants being one of them. For instance, by placing the coordinates of points in a determinant as shown in the exercise, collinearity can be established effectively. It is the mathematical rigor provided by algebra that allows for such precise interpretations based on matrix determinants.

Real-world Applications

Understanding collinearity is not only academically useful but also critical in fields like navigational systems, computer graphics, and data analysis where linear relationships are important. In linear regression, for example, collinearity among variables can significantly affect the accuracy of a predictive model.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to describe the position of points on the plane. This field involves plotting points, lines, curves, and shapes using a set of coordinates—usually, in two dimensions, these coordinates are (x, y).

One of the core principles of coordinate geometry involves understanding the relationships between points. For instance, to determine whether points are collinear, various methods like slope comparison and distance formula can be used. However, using a matrix determinant streamlines this process by transforming geometric conditions into an algebraic calculation. When points are not collinear, coordinate geometry helps to define the shape and size of the triangle they form and enables calculations of other geometric properties like area and perimeter.

Linking Algebra and Geometry

The amalgamation of algebraic concepts, like determinants, with geometric contexts exemplifies the power of coordinate geometry. It provides a versatile toolkit for solving geometrical problems algebraically, and vice versa, enhancing the understanding and application of these seemingly distinct areas of mathematics.