Question

An equation of the line containing the two points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) may be expressed as the determinant $$ \left|\begin{array}{lll} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{array}\right|=0 $$ Prove this result by expanding the determinant and comparing the result to the two-point form of the equation of a line.

Short answer

Determinant expansion matches the two-point line equation form.

Step-by-step solution

- Write the determinant form

The equation of the line containing the points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) is given by the determinant expression: \[ \left|\begin{array}{lll} x & y & 1 \ x_{1} & y_{1} & 1 \ x_{2} & y_{2} & 1 \end{array}\right|=0 \]

- Expand the determinant

Expand the determinant along the first row: \[ \left|\begin{array}{ccc} x & y & 1 \ x_1 & y_1 & 1 \ x_2 & y_2 & 1 \end{array}\right| = x \left|\begin{array}{cc} y_1 & 1 \ y_2 & 1 \end{array}\right| - y \left|\begin{array}{cc} x_1 & 1 \ x_2 & 1 \end{array}\right| + 1 \left|\begin{array}{cc} x_1 & y_1 \ x_2 & y_2 \end{array}\right| \]

- Calculate the 2x2 determinants

Evaluate each of the 2x2 determinants: \[ x( y_1 \cdot 1 - 1 \cdot y_2 ) - y( x_1 \cdot 1 - 1 \cdot x_2 ) + 1 ( x_1 \cdot y_2 - y_1 \cdot x_2 ) = x(y_1 - y_2) - y(x_1 - x_2) + (x_1 y_2 - y_1 x_2) \]

- Combine the terms

Combine the terms to form the equation: \[ x (y_1 - y_2) - y(x_1 - x_2) + x_1 y_2 - y_1 x_2 = 0 \]

- Compare with the two-point form

Recall the two-point form of the line equation: \[ \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} \] Rearrange this form to match the derived equation: \[ (y - y_1)(x_2 - x_1) = (x - x_1)(y_2 - y_1) \] Expanding and simplifying, we get: \[ y x_2 - y y_1 - y_1 x_2 + y_1 x_1 = x y_2 - x y_1 - x_1 y_2 + x_1 y_1 \] Which simplifies to: \[ x(y_1 - y_2) - y(x_1 - x_2) + x_1 y_2 - y_1 x_2 = 0 \] This matches the expression from the determinant.

Key concepts

Two-Point Form of a Line

The two-point form of a line helps us derive the equation of a straight line if two points on the line are known. Given two points \((x_1, y_1)\) and \((x_2, y_2)\), the equation of the line passing through them can be written as:
\(\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}\).
This means that the difference in \(y\) coordinates over the difference in \(x\) coordinates must be constant for every point on the line.
By rearranging, this can also be written as:
\((y - y_1)(x_2 - x_1) = (x - x_1)(y_2 - y_1)\). This form is useful when you want to look at the relationship between any point on a line and two known points on the same line.

Expanding a Determinant

A determinant is a special number calculated from a matrix. To find the equation of a line using determinants, we use the given points and write the determinant as shown:
\(\begin{vmatrix} x & y & 1 \ x_1 & y_1 & 1 \ x_2 & y_2 & 1 \end{vmatrix} = 0\).
Expanding the determinant down the first row, we break it down into smaller parts:
\(\begin{vmatrix} x & y & 1 \ x_1 & y_1 & 1 \ x_2 & y_2 & 1 \end{vmatrix} = x \begin{vmatrix} y_1 & 1 \ y_2 & 1 \end{vmatrix} - y \begin{vmatrix} x_1 & 1 \ x_2 & 1 \end{vmatrix} + 1 \begin{vmatrix} x_1 & y_1 \ x_2 & y_2 \end{vmatrix}\).
Then, we calculate the determinant of each \(2x2\) matrix:
\(= x(y_1 \cdot 1 - 1 \cdot y_2) - y(x_1 \cdot 1 - 1 \cdot x_2) + 1(x_1 \cdot y_2 - y_1 \cdot x_2)\).
This simplifies to:
\(x(y_1 - y_2) - y(x_1 - x_2) + (x_1 y_2 - y_1 x_2)\).
Understanding how to expand and simplify determinants is crucial for solving linear equations.

Equation of a Line

The equation of a line represents all of the points that lie on a line. From the previous steps, we derived the equation of the line in determinant form and expanded it to arrive at: \( x(y_1 - y_2) - y(x_1 - x_2) + x_1 y_2 - y_1 x_2 = 0 \).
This can be compared to the two-point form of the line, which is: \( (y - y_1)(x_2 - x_1) = (x - x_1)(y_2 - y_1) \).
By expanding this, we get:
\( y x_2 - y y_1 - y_1 x_2 + y_1 x_1 = x y_2 - x y_1 - x_1 y_2 + x_1 y_1 \),
which simplifies to: \( x(y_1 - y_2) - y(x_1 - x_2) + x_1 y_2 - y_1 x_2 = 0 \).
Therefore, both the determinant method and the algebraic method give the same equation, verifying their consistency. Having these forms is important for analyzing and understanding the properties of straight lines in coordinate geometry.