Question
Find the magnitude of vector v, that has an initial point of (1, 3) and a terminal point of (7, 1)
2
10
2
14
4
2
4
5
Answers
Answered by
GPT-4o mini
To find the magnitude of the vector \( \mathbf{v} \) given its initial and terminal points, we first determine the components of the vector.
The initial point is \( (1, 3) \) and the terminal point is \( (7, 1) \).
The components of vector \( \mathbf{v} \) can be calculated as follows:
\[
\mathbf{v} = (x_2 - x_1, y_2 - y_1) = (7 - 1, 1 - 3) = (6, -2)
\]
Next, we compute the magnitude of vector \( \mathbf{v} \) using the formula for the magnitude of a vector \( \mathbf{v} = (a, b) \):
\[
\|\mathbf{v}\| = \sqrt{a^2 + b^2}
\]
Substituting in the components:
\[
\|\mathbf{v}\| = \sqrt{6^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10}
\]
Thus, the magnitude of vector \( \mathbf{v} \) is \( 2\sqrt{10} \).
Among the provided options, \( 2\sqrt{10} \) is not explicitly listed, but can be approximated for context. However, if we strictly consider the integer or simple factors, none of the given answers match \( 2\sqrt{10} \).
If the options given are the only choices, you might want to double-check the problem or see if any simplifications or approximations were intended.
The initial point is \( (1, 3) \) and the terminal point is \( (7, 1) \).
The components of vector \( \mathbf{v} \) can be calculated as follows:
\[
\mathbf{v} = (x_2 - x_1, y_2 - y_1) = (7 - 1, 1 - 3) = (6, -2)
\]
Next, we compute the magnitude of vector \( \mathbf{v} \) using the formula for the magnitude of a vector \( \mathbf{v} = (a, b) \):
\[
\|\mathbf{v}\| = \sqrt{a^2 + b^2}
\]
Substituting in the components:
\[
\|\mathbf{v}\| = \sqrt{6^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10}
\]
Thus, the magnitude of vector \( \mathbf{v} \) is \( 2\sqrt{10} \).
Among the provided options, \( 2\sqrt{10} \) is not explicitly listed, but can be approximated for context. However, if we strictly consider the integer or simple factors, none of the given answers match \( 2\sqrt{10} \).
If the options given are the only choices, you might want to double-check the problem or see if any simplifications or approximations were intended.