Question

A graph of three vectors is shown.The graph shows vector v with initial point left-parenthesis negative 1 comma 0 right-parenthesis and terminal point left-parenthesis 3 comma 3 right-parenthesis, vector w with initial point left-parenthesis 3 comma 3 right-parenthesis and terminal point left-parenthesis 5 comma 2 right-parenthesis, and vector u with initial point left-parenthesis negative 1 comma 0 right-parenthesis and terminal point left-parenthesis 5 comma 2 right-parenthesis.
Question
Use the graph to answer the question.
Which statement matches the vector operation shown on the coordinate grid?
(1 point)
Responses

v + u = w for v = 4i + 3j, u = 6i + 6j, and bold lower w equals 2 bold lower i minus bold lower j
v + u = w for v = 4 i + 3 j , u = 6 i + 6 j, and Image with alt text: bold lower w equals 2 bold lower i minus bold lower j

v + u = w for v = 4i + 3j, u = 6i + 2j, and w = 8i + 5j
v + u = w for v = 4 i + 3 j , u = 6 i + 2 j , and w = 8 i + 5 j

v + w = u for v = 4i + 3j, bold lower w equals 2 bold lower i minus bold lower j, and u = 6i + 2j
v + w = u for v = 4 i + 3 j , Image with alt text: bold lower w equals 2 bold lower i minus bold lower j , and u = 6 i + 2 j

v + w = u for v = 4i + 3j, bold lower w equals 2 bold lower i minus bold lower j, and u = 2i + 4j

Answer 1

To determine the correct statement that matches the vector operation represented in the graph, we first need to analyze the vectors based on their initial and terminal points:

1. **Vector \( v \)** begins at point \((-1, 0)\) and ends at point \((3, 3)\):
- The components of vector \( v \):
\[
v = (3 - (-1), 3 - 0) = (4, 3)
\]
- In terms of unit vectors, this can be written as:
\[
v = 4i + 3j
\]

2. **Vector \( w \)** begins at point \((3, 3)\) and ends at point \((5, 2)\):
- The components of vector \( w \):
\[
w = (5 - 3, 2 - 3) = (2, -1)
\]
- In terms of unit vectors, this can be written as:
\[
w = 2i - j
\]

3. **Vector \( u \)** starts at point \((-1, 0)\) and ends at point \((5, 2)\):
- The components of vector \( u \):
\[
u = (5 - (-1), 2 - 0) = (6, 2)
\]
- In terms of unit vectors, this can be written as:
\[
u = 6i + 2j
\]

Now we can check the statements:

- The operation \( v + u \) computes as follows:
\[
v + u = (4i + 3j) + (6i + 2j) = (4 + 6)i + (3 + 2)j = 10i + 5j
\]

- To find the equivalent vector representation of \( w \):
\[
w = 2i - j \quad \text{(based on our earlier calculation)}
\]

Now, let's review each of the provided statements:

1. \( v + u = w \leftarrow \text{False} \) (Since \( v + u \) gives a different vector)
2. \( v + u = w \leftarrow \text{False} \)
3. \( v + w = u \leftarrow \text{True} \) (needs to be checked)
4. \( v + w = u \leftarrow \text{False} \)

Now check \( v + w \):
\[
v + w = (4i + 3j) + (2i - j) = (4 + 2)i + (3 - 1)j = 6i + 2j
\]

Since we already established that \( u = 6i + 2j \), we can conclude that:
\[
v + w = u
\]

Thus, the correct option that matches the vector operation shown on the coordinate grid is:
**v + w = u for \( v = 4i + 3j \), \( w = 2i - j \), and \( u = 6i + 2j \)** (This is the correct response).