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EXERCISE 1.1 Given the three points Po = (1, 1), P₁ = (2, 2.5), and P₂ = (3, 4), are they collinear?

EXERCISE 1.1 Given the three points Po = (1, 1), P₁ = (2, 2.5), and P₂ = (3, 4), are they collinear?

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### Exercise 1.1: Collinearity of Points **Problem Statement:** Given the three points \(P_0 = (1, 1)\), \(P_1 = (2, 2.5)\), and \(P_2 = (3, 4)\), are they collinear? **Hint:** Try to represent one point as the weighted sum of the other two points. **Objective:** To determine whether the given points lie on a single straight line. **Approach:** - Collinearity of three points can be checked by determining if the area of the triangle formed by these points is zero. - Another method is to use the concept of the weighted sum of points. You can try expressing one of the points as a linear combination of the other two points to check for collinearity. In detail: - Representing P1 as a weighted sum of P0 and P2 - Solve for weights if possible - If valid weights exist, points are collinear; otherwise, they are not. By following this approach, you can determine whether the mentioned points are collinear. **Remember:** Collinear points lie on the same straight line, thus having a determinant of zero for the area formulated by the triangle of these points or fitting into the linear combination representation. Feel free to experiment and derive proofs using various approaches and mathematical techniques for a deeper understanding of collinearity.
Transcribed Image Text:### Exercise 1.1: Collinearity of Points **Problem Statement:** Given the three points \(P_0 = (1, 1)\), \(P_1 = (2, 2.5)\), and \(P_2 = (3, 4)\), are they collinear? **Hint:** Try to represent one point as the weighted sum of the other two points. **Objective:** To determine whether the given points lie on a single straight line. **Approach:** - Collinearity of three points can be checked by determining if the area of the triangle formed by these points is zero. - Another method is to use the concept of the weighted sum of points. You can try expressing one of the points as a linear combination of the other two points to check for collinearity. In detail: - Representing P1 as a weighted sum of P0 and P2 - Solve for weights if possible - If valid weights exist, points are collinear; otherwise, they are not. By following this approach, you can determine whether the mentioned points are collinear. **Remember:** Collinear points lie on the same straight line, thus having a determinant of zero for the area formulated by the triangle of these points or fitting into the linear combination representation. Feel free to experiment and derive proofs using various approaches and mathematical techniques for a deeper understanding of collinearity.
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