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:
-
Point \( (-0.2, 0.4) \): \[ (-2 - (-0.2))^2 + (4 - 0.4)^2 = (-2 + 0.2)^2 + (3.6)^2 = (-1.8)^2 + (3.6)^2 = 3.24 + 12.96 = 16.2 \quad \text{(not equal to 100)} \]
-
Point \( (-8, -4) \): \[ (-2 - (-8))^2 + (4 - (-4))^2 = (-2 + 8)^2 + (4 + 4)^2 = (6)^2 + (8)^2 = 36 + 64 = 100 \quad \text{(equal to 100)} \]
-
Point \( (-12, -6) \): \[ (-2 - (-12))^2 + (4 - (-6))^2 = (-2 + 12)^2 + (4 + 6)^2 = (10)^2 + (10)^2 = 100 + 100 = 200 \quad \text{(not equal to 100)} \]
-
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 \( (-8, -4) \).