Problem 64 - Precalculus Mathematics for Calculus

Question

(a) If three points lie on a line, what is the area of the "triangle" that they determine? Use the answer to this question, together with the determinant formula for the area of a triangle, to explain why the points \(\left(a_{1}, b_{1}\right)\) \(\left(a_{2}, b_{2}\right),\) and \(\left(a_{3}, b_{3}\right)\) are collinear if and only if $$\left|\begin{array}{lll} a_{1} & b_{1} & 1 \\ a_{2} & b_{2} & 1 \\ a_{3} & b_{3} & 1 \end{array}\right|=0$$ (b) Use a determinant to check whether each set of points is collinear. Graph them to verify your answer. (I) \((-6,4),(2,10),(6,13)\) (II) \((-5,10),(2,6),(15,-2)\)

Step-by-Step Solution

Answer

(I) Points are collinear. (II) Points are not collinear.

Step 1: Understand the Concept of Collinearity

When three points are collinear, they lie on the same straight line. This means that they do not form a closed area such as that of a triangle. Hence, the 'triangle' formed by collinear points has an area of zero.

Step 2: Formula for Area of a Triangle Using Determinants

The area of a triangle formed by points \( (a_1, b_1), (a_2, b_2), (a_3, b_3) \) can be calculated using the determinant formula: \[ \text{Area} = \frac{1}{2} \left| \begin{array}{ccc} a_1 & b_1 & 1 \ a_2 & b_2 & 1 \ a_3 & b_3 & 1 \end{array} \right| \]If the points are collinear, this determinant equals zero, leading to zero area.

Step 3: Solve Part (a) of the Exercise

Given the formula for the area of a triangle using a determinant, we see that if the absolute value of the determinant equals zero, the points are collinear. Thus, \[ \left| \begin{array}{ccc} a_1 & b_1 & 1 \ a_2 & b_2 & 1 \ a_3 & b_3 & 1 \end{array} \right| = 0 \]indicates that the points \( (a_1, b_1), (a_2, b_2), (a_3, b_3) \) are collinear.

Step 4: Part (b) - Point Set (I) Verification

Check collinearity for points \( (-6,4), (2,10), (6,13) \):The determinant is simple to calculate:\[ \left| \begin{array}{ccc} -6 & 4 & 1 \ 2 & 10 & 1 \ 6 & 13 & 1 \end{array} \right| = -6(10-13) - 4(2-6) + (2 \cdot 13 - 10 \cdot 6). \]Compute step-by-step:- \( -6 \times -3 = 18 \)- \( -4 \times -4 = 16 \)- \( (26 - 60) = -34 \)Sum: \(\ 18 + 16 - 34 = 0\)Since it equals zero, the points are collinear.

Step 5: Part (b) - Point Set (II) Verification

Check for points \( (-5,10), (2,6), (15,-2) \):Calculate the determinant:\[ \left| \begin{array}{ccc} -5 & 10 & 1 \ 2 & 6 & 1 \ 15 & -2 & 1 \end{array} \right| = -5(6 - (-2)) - 10(2 - 15) + (2(-2) - 6 \cdot 15). \]Step-by-step:- \( -5 \times 8 = -40 \)- \( -10 \times -13 = 130 \)- \( -4 - 90 = -94 \)Sum: \( -40 - 130 - 94 = -264 \)Since it does not equal zero, the points are not collinear.

Step 6: Graphical Verification

Plot each set of points on a coordinate grid to visually verify - For point set (I): The points lie on a straight line, confirming collinearity. - For point set (II): The points do not lie on a straight line, confirming non-collinearity.

Key Concepts

Collinear Points

Collinear points are points that lie on the same straight line. In simple terms, three or more points are collinear if we can connect them with a single straight line without picking up the pen. Importantly, collinear points do not form a triangle since they do not enclose any area. This means that the area formed by collinear points is always zero.

Understanding collinearity is crucial when studying coordinate geometry because it tells us about the geometric relationship between points. If you encounter three points in an exam or homework, you can quickly check if they're collinear by calculating the area of the triangle they might form. If the area is zero, the points are collinear. This concept is beautifully encapsulated using the determinant formula for the area of a triangle.

Area of a Triangle

The area of a triangle in coordinate geometry can be elegantly calculated using determinants. If you have three points, let's say \((a_1, b_1), (a_2, b_2), (a_3, b_3)\), you can find the area by plugging these coordinates into the following formula:\[\text{Area} = \frac{1}{2} \left| \begin{array}{ccc} a_1 & b_1 & 1 \ a_2 & b_2 & 1 \ a_3 & b_3 & 1 \end{array} \right|\]This determinant method is highly effective in simplifying the process of finding the area, especially when you have coordinates at hand. If you do the math and find that the determinant is zero, then your triangle has no area because the points are collinear. This is a pivotal concept not only for solving problems but also for checking the validity of geometric properties in various mathematical contexts.

Coordinate Geometry

Coordinate geometry, or analytic geometry, allows us to use algebra to study geometric problems. This involves points, lines, curves, and other shapes using a coordinate system, often the Cartesian plane.

A key part of coordinate geometry is understanding how to represent geometric shapes using equations and coordinates. Using coordinates, you can easily determine relationships like whether points are collinear or the area enclosed by those points.

• Collinear Points: Use the determinant of the coordinates to check if points lie on the same line.
• Area Measurement: Calculate triangle areas using coordinates, where the determinant again plays a major role.
• Graphing: Visualizing solutions by plotting points can confirm correct calculations.

In essence, coordinate geometry transforms complex geometric problems into algebraic challenges, making them easier to solve and verify using mathematical techniques.