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

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
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 = 2 i + 4 j
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Answer 1

To determine which statement matches the vector operation shown on the coordinate grid, we first need to calculate the vectors \( v \), \( w \), and \( u \) based on their endpoints.

1. Vector \( v \):

   • Initial point: \( (-1, 0) \)
   • Terminal point: \( (3, 3) \)
   • Components:
     • \( v = (3 - (-1))i + (3 - 0)j = 4i + 3j \)

2. Vector \( w \):

   • Initial point: \( (3, 3) \)
   • Terminal point: \( (5, 2) \)
   • Components:
     • \( w = (5 - 3)i + (2 - 3)j = 2i - j \)

3. Vector \( u \):

   • Initial point: \( (-1, 0) \)
   • Terminal point: \( (5, 2) \)
   • Components:
     • \( u = (5 - (-1))i + (2 - 0)j = 6i + 2j \)

Now that we have the vectors:

• \( v = 4i + 3j \)
• \( w = 2i - j \)
• \( u = 6i + 2j \)

Next, we can check the vector operations to see if they sum up correctly:

• Check \( v + u\): \[ v + u = (4i + 3j) + (6i + 2j) = (4 + 6)i + (3 + 2)j = 10i + 5j \] Therefore, this does NOT match \( w \) since \( w = 2i - j \).

• Check \( v + w \): \[ v + w = (4i + 3j) + (2i - j) = (4 + 2)i + (3 - 1)j = 6i + 2j \] This gives us \( u \), confirming that \( v + w = u \).

Given these calculations, we evaluate the answer options:

• The correct statement from the options is:
  \( v + w = u \) for \( v = 4i + 3j \), \( w = 2i - j \), and \( u = 6i + 2j \).

Thus, the response that matches the vector operations shown is: v + w = u for v = 4i + 3j, w = 2i - j, and u = 6i + 2j.