What is the value of the vectors AB, BC, and CA when A, B, and C are the vertices of a triangle?

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To find the values of the vectors AB, BC, and CA for the triangle formed by the vertices A, B, and C, we first need to understand what these vectors represent.

If we denote the vertices of the triangle as:

  • A(x1, y1)
  • B(x2, y2)
  • C(x3, y3)

Then the vectors can be expressed as follows:

  • Vector AB: This vector goes from point A to point B. It can be calculated by subtracting the coordinates of A from those of B:
  • AB = B – A = (x2 – x1, y2 – y1)
  • Vector BC: This vector goes from point B to point C:
  • BC = C – B = (x3 – x2, y3 – y2)
  • Vector CA: This vector goes from point C to point A:
  • CA = A – C = (x1 – x3, y1 – y3)

Now, putting these vectors together, you get:

  • AB = (x2 – x1, y2 – y1)
  • BC = (x3 – x2, y3 – y2)
  • CA = (x1 – x3, y1 – y3)

To analyze the triangle’s properties further, you can calculate the magnitudes of these vectors to explore the lengths of the sides of the triangle:

  • Magnitude of vector AB: |AB| = √((x2 – x1)2 + (y2 – y1)2)
  • Magnitude of vector BC: |BC| = √((x3 – x2)2 + (y3 – y2)2)
  • Magnitude of vector CA: |CA| = √((x1 – x3)2 + (y1 – y3)2)

This step clearly shows how to find the values for vectors AB, BC, and CA using simple coordinate subtraction. Understanding these vectors is crucial in vector calculus, physics, and various engineering domains.

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