Problem 3 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Using determinants, show that the points \((3,8),(-4,2)\) and \((10,14)\) are collinear.

Step-by-Step Solution

Answer

The points are collinear because the determinant is 0.

Step 1: Understand Collinearity with Determinants

To show that three points are collinear, we can use the determinant method. For points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\), the determinant of the matrix \(\begin{vmatrix} 1 & x_1 & y_1 \1 & x_2 & y_2 \1 & x_3 & y_3 \end{vmatrix} = 0 \)\ if the points are collinear.

Step 2: Write the Matrix

Place the points \((3,8), (-4,2), (10,14)\) into the matrix as follows:\[\begin{vmatrix} 1 & 3 & 8 \1 & -4 & 2 \1 & 10 & 14 \end{vmatrix}\]

Step 3: Calculate the Determinant

Calculate the determinant of the matrix. The formula for a 3x3 determinant is:\[\begin{vmatrix}a & b & c \d & e & f \g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)\]Substitute \(a=1\), \(b=3\), \(c=8\), \(d=1\), \(e=-4\), \(f=2\), \(g=1\), \(h=10\), \(i=14\) into the formula.

Step 4: Substitute Values and Simplify

Substitute the values into the determinant formula:\[1(((-4)(14) - (2)(10)) - 3((1)(14) - (2)(1)) + 8((1)(10) - ((-4)(1)) = 0\]Simplify each part:1. \((-4)(14) - (2)(10) = -56 - 20 = -76\)2. \((1)(14) - (2)(1) = 14 - 2 = 12\)3. \((1)(10) - ((-4)(1)) = 10 + 4 = 14\)Substitute back:\[1(-76) - 3(12) + 8(14) = 0\]

Step 5: Perform the Final Calculations

Calculate each term:1. \(1(-76) = -76\)2. \(-3(12) = -36\)3. \(+8(14) = 112\)Combine: \(-76 - 36 + 112 = 0\).

Step 6: Conclusion on Collinearity

Since the determinant value is 0, the points \((3,8)\), \((-4,2)\), and \((10,14)\) are collinear.

Key Concepts

Collinearity

Collinearity is a term used in geometry and algebra to describe a scenario where three or more points lie on a single straight line. Verifying collinearity helps in understanding the relationship between these points in the coordinate plane. This can often be shown through algebraic methods, one of which employs determinants. A determinant is a calculated value from a square matrix that provides insight into the matrix's properties, including whether a set of points defined by the matrix are collinear. When using a determinant to check for collinearity, we set up a 3x3 matrix using the coordinates of the three points in question. By calculating the determinant of this matrix, we determine if the points are collinear; if the determinant equals zero, the points lie on a straight line. This is because a zero determinant indicates that the area of the triangle formed by the points is zero, implying that the points do not form a triangle and are thus collinear.

Matrix

A matrix is a rectangular array of numbers arranged in rows and columns. In mathematics, matrices are used to represent and solve linear equations, transformations, and more. A matrix is particularly useful in various fields, including computer graphics, physics, and statistics, because it can compactly represent data and operations. To check for collinearity using matrices, we organize the coordinates of the points into a 3x3 matrix with rows where each consists of:

• A 1 (for consistency across the operations)
• The x-coordinate
• The y-coordinate

This structure allows us to easily apply the determinant formula, providing a straightforward method to analyze relationships between points. Matrices simplify the mathematical operations needed in determining properties like collinearity, enabling us to deploy algebraic techniques efficiently.

3x3 Determinant Formula

The determinant of a 3x3 matrix is a specific value calculated using a standard formula that involves all the elements of the matrix. This calculation is crucial for problems involving linear transformations and collinearity checks among others.For a 3x3 matrix of the form:\[ \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} \]the determinant, denoted as \( \text{det}(A) \), is calculated with the expression:\[ a(ei - fh) - b(di - fg) + c(dh - eg) \]Each of the terms in the formula results from the expansion using minors, a method that simplifies the computation by breaking it down into smaller, manageable 2x2 determinants. Substituting the coordinates of the points into this formula allows us to compute the determinant value, which indicates whether the involved points are collinear. Hence, if you find the determinant to be zero, it confirms the collinearity of the points.