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Vector u is represented by the directed line segment Modifying above upper R upper S with right arrow and vector v is represented by the directed line segment Modifying above upper O upper P with right arrow. If upper R equals left parenthesis 8 comma negative 2 right parenthesis, upper S equals left parenthesis 11 comma negative 6 right parenthesis, upper O equals left parenthesis negative 3 comma negative 9 right parenthesis, and upper P equals left parenthesis 0 comma negative 13 right parenthesis, are u and v equal? Explain how you reached your answer.

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To determine if the two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are equal, we first need to compute their components based on the given points.

Step 1: Calculate Vector \( \mathbf{u} \)

Vector \( \mathbf{u} \) is represented by the directed line segment from point \( R \) to point \( S \). The coordinates of \( R \) and \( S \) are given as:

  • \( R = (8, -2) \)
  • \( S = (11, -6) \)

The components of vector \( \mathbf{u} \) are calculated as follows: \[ \mathbf{u} = S - R = (11 - 8, -6 - (-2)) = (3, -4) \]

Step 2: Calculate Vector \( \mathbf{v} \)

Vector \( \mathbf{v} \) is represented by the directed line segment from point \( O \) to point \( P \). The coordinates of \( O \) and \( P \) are given as:

  • \( O = (-3, -9) \)
  • \( P = (0, -13) \)

The components of vector \( \mathbf{v} \) are calculated as follows: \[ \mathbf{v} = P - O = (0 - (-3), -13 - (-9)) = (3, -4) \]

Step 3: Compare the Vectors

We find that: \[ \mathbf{u} = (3, -4) \quad \text{and} \quad \mathbf{v} = (3, -4) \]

Since the components of both vectors \( \mathbf{u} \) and \( \mathbf{v} \) are the same, we conclude that: \[ \mathbf{u} = \mathbf{v} \]

Conclusion

The vectors \( \mathbf{u} \) and \( \mathbf{v} \) are equal because they have the same components \( (3, -4) \). Thus, \( \mathbf{u} = \mathbf{v} \).