Problem 66 - Precalculus

Question

66\. Geometry. If three points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) are collinear (lie on the same line), then the following determinant equation must be satisfied: $$\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\\x_{3} & y_{3} & 1 \end{array}\right|=0$$ Determine whether \((0,5),(2,0),\) and (1,2)

Step-by-Step Solution

Answer

The points (0,5), (2,0), and (1,2) are not collinear because the determinant is -1, not 0.

Step 1: Write down the matrix for the points

We are given the points \((x_1, y_1) = (0, 5)\), \((x_2, y_2) = (2, 0)\), and \((x_3, y_3) = (1, 2)\). The matrix for these points is: \[\begin{bmatrix}0 & 5 & 1 \2 & 0 & 1 \1 & 2 & 1\end{bmatrix}\]

Step 2: Calculate the determinant of the matrix

The determinant of a 3x3 matrix \(\begin{bmatrix}a & b & c \d & e & f \g & h & i\end{bmatrix}\) is given by \(a(ei − fh) − b(di − fg) + c(dh − eg)\).For the given matrix:\[\begin{vmatrix}0 & 5 & 1 \ 2 & 0 & 1 \ 1 & 2 & 1\end{vmatrix} = 0(0\cdot1 - 1\cdot2) - 5(2\cdot1 - 1\cdot1) + 1(2\cdot2 - 0\cdot1)\]

Step 3: Simplify the determinant

Simplify the expression from Step 2: \[0(0 - 2) - 5(2 - 1) + 1(4 - 0)\]This simplifies to: \[0 - 5(1) + 4 = 0 - 5 + 4 = -1\]

Step 4: Determine if the determinant equals zero

After calculating the determinant, we found it to be \(-1\). Since it is not zero, the given points \((0,5), (2,0),\) and \((1,2)\) are not collinear.

Key Concepts

Determinant

A determinant is a special number that can be calculated from a square matrix. It's an important concept in linear algebra with significant applications in geometry, particularly for determining specific attributes of geometric figures. In the 2x2 case, the determinant of matrix \[\begin{bmatrix}a & b \c & d\end{bmatrix}\]is calculated as \( ad - bc \). For a 3x3 matrix, the calculation is slightly more complex as it involves inputs from all elements of the matrix. Calculating the determinant of a 3x3 matrix can act like a test for various properties like the invertibility of the matrix or collinearity of points in a geometric setting. If the determinant is zero, it indicates a special scenario: that the columns (or rows) of the matrix are linearly dependent.

3x3 Matrix

A 3x3 matrix is a square matrix that has three rows and three columns. It's used in various fields, including computer graphics, physics, and mathematics, particularly in matrix algebra and geometry. The matrix mentioned in the exercise has this form:\[\begin{bmatrix}x_{1} & y_{1} & 1 \x_{2} & y_{2} & 1 \x_{3} & y_{3} & 1\end{bmatrix}\]Each entry \((x, y)\) represents the coordinates of a point in the 2D plane, and the column of ones is added to facilitate the usage of matrix determinants in determining collinearity. Incorporating the third column filled with ones converts the 2D points into the 3D plane, which is essential for verifying whether the points are collinear by finding if the determinant is zero.

Matrix Algebra

Matrix algebra encompasses a variety of operations, including addition, subtraction, multiplication, and determining the inverse or transpose of matrices. One of the pivotal operations in matrix algebra is determining the determinant of a matrix. For 3x3 matrices, the determinant is calculated using all elements of the matrix: - Select a row or column. - Calculate three 2x2 determinants by eliminating one row and one column at a time. - Sum these products, adjusting for the sign (+ or -) based on the position in the original matrix. Performing these calculations helps to uncover various mathematical properties of the system or geometric properties of the represented points. Matrix algebra provides a systematic way to handle multiple linear equations and their geometrical interpretations.

Geometry

Geometry is the branch of mathematics that deals with shapes, congruence, and properties of space. In dealing with points in the plane, determining collinearity is a frequent geometric problem. Collinearity means all points lie on a single straight line. For three points, this can be tested using the determinant of the matrix composed of the points.
If you find the determinant of a matrix as zero, it implies the points are aligned perfectly in a line. This method is useful because it provides a numerical approach to assessing collinearity, bypassing visual or graphical assumptions that may be less precise. Therefore, using algebraic tools such as determinant calculations helps to solidify understanding and ensures accuracy in geometric judgments.