Question

Which of the following points could be the initial point of vector v if it has a magnitude of 10 and the terminal point left parenthesis negative 2 comma 4 right parenthesis?
(1 point)
Responses

left parenthesis negative 0.2 comma 0.4 right parenthesis
Image with alt text: left parenthesis negative 0.2 comma 0.4 right parenthesis

left parenthesis negative 8 comma negative 4 right parenthesis
Image with alt text: left parenthesis negative 8 comma negative 4 right parenthesis

left parenthesis negative 12 comma negative 6 right parenthesis
Image with alt text: left parenthesis negative 12 comma negative 6 right parenthesis

left parenthesis 1 comma 3 right parenthesis

Answer 1

To find the initial point of vector \( v \) with a given terminal point and magnitude, we can use the formula for the vector's length. The length \( |v| \) of the vector is determined by the coordinates of the terminal point \( (x_2, y_2) \) and the initial point \( (x_1, y_1) \):

\[
|v| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

In this case, the terminal point is \( (-2, 4) \) and the magnitude is \( 10 \). We need to find \( (x_1, y_1) \) such that:

\[
10 = \sqrt{((-2 - x_1)^2 + (4 - y_1)^2)}
\]

Squaring both sides gives:

\[
100 = (-2 - x_1)^2 + (4 - y_1)^2
\]

Next, we'll test each of the given points to see if they satisfy this equation:

1. **Point \( (-0.2, 0.4) \)**:
\[
(-2 - (-0.2))^2 + (4 - 0.4)^2 = (-2 + 0.2)^2 + (3.6)^2 = (-1.8)^2 + (3.6)^2 = 3.24 + 12.96 = 16.2 \quad \text{(not equal to 100)}
\]

2. **Point \( (-8, -4) \)**:
\[
(-2 - (-8))^2 + (4 - (-4))^2 = (-2 + 8)^2 + (4 + 4)^2 = (6)^2 + (8)^2 = 36 + 64 = 100 \quad \text{(equal to 100)}
\]

3. **Point \( (-12, -6) \)**:
\[
(-2 - (-12))^2 + (4 - (-6))^2 = (-2 + 12)^2 + (4 + 6)^2 = (10)^2 + (10)^2 = 100 + 100 = 200 \quad \text{(not equal to 100)}
\]

4. **Point \( (1, 3) \)**:
\[
(-2 - 1)^2 + (4 - 3)^2 = (-3)^2 + (1)^2 = 9 + 1 = 10 \quad \text{(not equal to 100)}
\]

After testing all points, the only point that yields the correct magnitude of the vector \( v \) is:

**Point \( (-8, -4) \)**.