Question

Show that the line through two distinct points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) in the plane has equation $$ \text { det }\left[\begin{array}{lll} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{array}\right]=0 $$

Short answer

The given determinant condition represents the standard line equation for collinear points in a plane.

Step-by-step solution

Explain the Determinant Setup

To find the equation of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the determinant setup, which is a way to express the condition that the third point \((x, y)\) lies on the same line. The matrix is constructed by arranging the coordinates: the first row for the general point \((x, y)\), the next two rows for the specific points \((x_1, y_1)\) and \((x_2, y_2)\). The determinant of this 3x3 matrix being zero indicates collinearity.

Compute the Determinant

Expand the determinant of the matrix: \[\text{det} \begin{bmatrix} x & y & 1 \ x_1 & y_1 & 1 \ x_2 & y_2 & 1 \end{bmatrix} = x(y_1 - y_2) - y(x_1 - x_2) + (x_1y_2 - x_2y_1)\]This gives us a linear equation in terms of \(x\) and \(y\), which is the equation of the line in the plane.

Interpret the Result

The expression \(x(y_1 - y_2) - y(x_1 - x_2) + (x_1y_2 - x_2y_1) = 0\) represents the equation of a line through the two given points. This is because if \(x, y\) satisfy the equation, they lie on the line formed by the points \((x_1, y_1)\) and \((x_2, y_2)\). The determinant approach showcases the condition of collinearity for these three points.

Key concepts

Determinant Method

The determinant method is a powerful tool in mathematics used to determine the collinearity of points, mainly in two-dimensional geometry. This method involves using a determinant, a special number that can be calculated from the elements of a square matrix.

To find the equation of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \), we start by setting up a \( 3 \times 3 \) matrix. This matrix includes the coordinates of the two given points and a general point \( (x, y) \) with an additional column of ones:

\[\text{Matrix} = \begin{bmatrix} x & y & 1 \ x_1 & y_1 & 1 x_2 & y_2 & 1 \end{bmatrix}\]

Calculating the determinant of this matrix provides an engaging way to verify if a third point lies on the same line as the given two points. If the determinant equals zero, it implies collinearity, showing that all three points align, thus helping derive the equation of the line.

Collinearity of Points

Collinearity is a geometric concept indicating that points lie on a single straight line. It's a crucial part of understanding linear relationships between points in a plane.

For the matrix setup mentioned, collinearity means that when we solve the determinant of the matrix:

\(\text{det} \begin{bmatrix} x & y & 1 \ x_1 & y_1 & 1 x_2 & y_2 & 1 \end{bmatrix} = 0\)

The result being zero shows us that the points \( (x,y) \), \( (x_1,y_1) \), and \( (x_2,y_2) \) are collinear. This means they all lie on the same straight line.

This approach is beneficial because it presents an algebraic method to solve what is typically a geometric problem. Instead of relying on visual alignment, we use a determinant to mathematically confirm the collinearity.

Matrix Representation in Geometry

Matrix representation is a versatile tool in geometry, allowing for the structured arrangement of coordinates to solve complex problems with systematic approaches. By using a matrix, one can apply algebraic techniques in geometric settings.

Consider the \( 3 \times 3 \) matrix configuration \(\begin{bmatrix} x & y & 1 \ x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ \end{bmatrix}\). This structure is not just about holding values; it serves an operational purpose.

The matrix allows us to explore the relationship between the points using determinant evaluation to determine collinearity. This utility makes it a bridge between algebra and geometry, bringing clarity to linear equations' roots.

Besides checking for collinear points, matrices serve many other purposes in geometry such as transformations and rotations, making them indispensable in advanced geometrical analysis.