Question

Vector w has a component form of left angle bracket negative 3 comma 5 right angle bracket. Which of the following could be the initial point A and the terminal point B of vector w?
(1 point)
Responses

upper A left parenthesis 2 comma 5 right parenthesis and upper B left parenthesis 6 comma 1 right parenthesis
Image with alt text: upper A left parenthesis 2 comma 5 right parenthesis and Image with alt text: upper B left parenthesis 6 comma 1 right parenthesis

upper A left parenthesis negative 1 comma 5 right parenthesis and upper B left parenthesis 4 comma 0 right parenthesis
Image with alt text: upper A left parenthesis negative 1 comma 5 right parenthesis and Image with alt text: upper B left parenthesis 4 comma 0 right parenthesis

upper A left parenthesis negative 1 comma 3 right parenthesis and upper B left parenthesis negative 2 comma 8 right parenthesis
Image with alt text: upper A left parenthesis negative 1 comma 3 right parenthesis and Image with alt text: upper B left parenthesis negative 2 comma 8 right parenthesis

upper A left parenthesis 6 comma negative 4 right parenthesis and upper B left parenthesis 3 comma 1 right parenthesis
Image with alt text: upper A left parenthesis 6 comma negative 4 right parenthesis and Image with alt text: upper B left parenthesis 3 comma 1 right parenthesis

Answer 1

To determine which pair of points \( A \) and \( B \) corresponds to the vector \( \mathbf{w} = \langle -3, 5 \rangle \), we can use the following relationship:

\[
\mathbf{w} = \overrightarrow{AB} = B - A
\]

If \( A \) has coordinates \( (a_1, a_2) \) and \( B \) has coordinates \( (b_1, b_2) \), then:

\[
\overrightarrow{AB} = \langle b_1 - a_1, b_2 - a_2 \rangle
\]

For \( \mathbf{w} = \langle -3, 5 \rangle \) we need:

\[
b_1 - a_1 = -3 \quad (1)
\]
\[
b_2 - a_2 = 5 \quad (2)
\]

Now, let's compute this for each option given:

1. **Option 1: \( A(2, 5) \) and \( B(6, 1) \)**
- \( b_1 - a_1 = 6 - 2 = 4 \) (not -3)
- \( b_2 - a_2 = 1 - 5 = -4 \) (not 5)

2. **Option 2: \( A(-1, 5) \) and \( B(4, 0) \)**
- \( b_1 - a_1 = 4 - (-1) = 4 + 1 = 5 \) (not -3)
- \( b_2 - a_2 = 0 - 5 = -5 \) (not 5)

3. **Option 3: \( A(-1, 3) \) and \( B(-2, 8) \)**
- \( b_1 - a_1 = -2 - (-1) = -2 + 1 = -1 \) (not -3)
- \( b_2 - a_2 = 8 - 3 = 5 \) (correct for second component)

4. **Option 4: \( A(6, -4) \) and \( B(3, 1) \)**
- \( b_1 - a_1 = 3 - 6 = -3 \) (correct for first component)
- \( b_2 - a_2 = 1 - (-4) = 1 + 4 = 5 \) (correct for second component)

Only **Option 4: \( A(6, -4) \) and \( B(3, 1) \)** satisfies both conditions:

\[
\overrightarrow{AB} = \langle -3, 5 \rangle
\]

Thus, the correct answer is:

**Option 4: \( A(6, -4) \) and \( B(3, 1) \)**.