Problem 21 - College Algebra and Calculus: An Applied Approach

Question

Use a determinant to determine whether the points are collinear. $$ (-2,-11),(4,13),(2,5) $$

Step-by-Step Solution

Answer

Calculate the result from the formula in Step 3. If that result is zero, then the points (-2,-11), (4,13), and (2,5) are collinear. If it is not zero, then they are not collinear.

Step 1: Forming the matrix

Put the coordinates of the points into a 3x3 matrix where the first column represents the x-coordinates, second column represents the y-coordinates, and the third column is filled with 1s. \[ \begin{bmatrix} -2 & -11 & 1 \\ 4 & 13 & 1 \\ 2 & 5 & 1 \end{bmatrix} \]

Step 2: Calculating the determinant

Calculate the determinant of the matrix. Use the formula for the determinant of a 3x3 matrix. \[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = aei + bfg + cdh - ceg - bdi - afh \] So, the determinant of our matrix is (-2*13*1) + (-11*1*2) + (1*4*5) - (1*13*2) - (-11*4*1) - (-2*1*5).

Step 3: Evaluating the determinant

Calculate the value from step 2. If the resultant value is zero, then the points are collinear. If it is not zero, then the points are not collinear.

Key Concepts

3x3 Matrix Determinant

Understanding the concept of a 3x3 matrix determinant is crucial for solving various problems in linear algebra, including determining the collinearity of points.

A determinant can be thought of as a scalar value that is a function of a square matrix. It reveals important properties about the matrix, such as whether it is invertible (a nonzero determinant) or singular (a zero determinant).

To calculate the determinant of a 3x3 matrix, we use a specific formula: \[ \begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} = aei + bfg + cdh - ceg - bdi - afh \] This formula essentially combines the products of the diagonals of the matrix in two directions – one from the top left to bottom right (adding those products together), and the other from the top right to bottom left (subtracting those products).

The calculation may first seem complicated, but by following an orderly approach, such as the one demonstrated in the problem, you can easily calculate the determinant step by step.

Collinearity Determination

The term 'collinear' in geometry refers to points that lie on the same straight line. To determine if points are collinear, algebraic methods like the 3x3 matrix determinant can be particularly helpful.

When we place the coordinates of the points into a 3x3 matrix with the third column filled with ones, this preparation sets the stage for determinant calculation. If the determinant of this matrix is zero, it indicates that the area of the triangle formed by these points is zero - implying that the points do not form a triangle at all, but rather lie on a straight line.

Mathematically, it's an elegant solution to an otherwise visual or graphical challenge. This method provides a definitive algebraic proof of collinearity which is essential in more complex geometric problems where visual determination is not practical or exact enough.

Algebraic Geometry

Algebraic geometry is the discipline that studies the connections between algebra, specifically algebraic equations, and geometric shapes. It's a rich and complex field that allows us to understand geometric properties using algebraic methods, such as the use of determinants for collinearity determination.

Within algebraic geometry, concepts like points, lines, and planes are not just visual or spatial entities; they are solutions to polynomial equations represented in algebraic form. The exercise demonstrates a practical application of algebraic geometry by algebraically solving a problem that traditionally relies on geometric intuition.

Through algebraic geometry, one can generalize the collinearity of points to more complex situations, like the intersection of higher-dimensional shapes, and use similar algebraic tools to explore these concepts. It’s a fascinating intersection of mathematical disciplines that extends far beyond the straightforward problem we’ve addressed with determinants.