Question

Question
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.


(4 points)

Answer 1

To determine whether the vectors \( \mathbf{u} \) (represented by the directed line segment from \( R \) to \( S \)) and \( \mathbf{v} \) (represented by the directed line segment from \( O \) to \( P \)) are equal, we need to find the components of each vector.

1. **Calculate vector \( \mathbf{u} \)** from \( R \) to \( S \):

Given points:
\[
R = (8, -2)
\]
\[
S = (11, -6)
\]

The components of vector \( \mathbf{u} \) can be found using the formula:
\[
\mathbf{u} = \langle S_x - R_x, S_y - R_y \rangle = \langle 11 - 8, -6 - (-2) \rangle
\]

Calculating the components:
\[
\mathbf{u} = \langle 3, -4 \rangle
\]

2. **Calculate vector \( \mathbf{v} \)** from \( O \) to \( P \):

Given points:
\[
O = (-3, -9)
\]
\[
P = (0, -13)
\]

The components of vector \( \mathbf{v} \) can be found similarly:
\[
\mathbf{v} = \langle P_x - O_x, P_y - O_y \rangle = \langle 0 - (-3), -13 - (-9) \rangle
\]

Calculating the components:
\[
\mathbf{v} = \langle 3, -4 \rangle
\]

3. **Compare vectors \( \mathbf{u} \) and \( \mathbf{v} \)**:

From the calculations:
\[
\mathbf{u} = \langle 3, -4 \rangle
\]
\[
\mathbf{v} = \langle 3, -4 \rangle
\]

Since both vectors have the same components, we conclude that:

**Answer**: Yes, the vectors \( \mathbf{u} \) and \( \mathbf{v} \) are equal because they have the same components of \( \langle 3, -4 \rangle \).