Question
Find the magnitude of vector v, that has an initial point of (1, 3) and a terminal point of (7, 1)
Answers
Answered by
GPT-4o mini
To find the magnitude of the vector \( \mathbf{v} \) with initial point \( (1, 3) \) and terminal point \( (7, 1) \), we first need to determine the components of the vector.
The components of the vector \( \mathbf{v} \) can be computed as follows:
\[
\mathbf{v} = (x_2 - x_1, y_2 - y_1) = (7 - 1, 1 - 3) = (6, -2)
\]
Next, we calculate the magnitude (length) of the vector using the formula:
\[
|\mathbf{v}| = \sqrt{(x^2 + y^2)}
\]
Plugging in our components \( (6, -2) \):
\[
|\mathbf{v}| = \sqrt{(6^2 + (-2)^2)} = \sqrt{(36 + 4)} = \sqrt{40}
\]
The magnitude can be simplified further:
\[
|\mathbf{v}| = \sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}
\]
Thus, the magnitude of the vector \( \mathbf{v} \) is
\[
\boxed{2\sqrt{10}}
\]
The components of the vector \( \mathbf{v} \) can be computed as follows:
\[
\mathbf{v} = (x_2 - x_1, y_2 - y_1) = (7 - 1, 1 - 3) = (6, -2)
\]
Next, we calculate the magnitude (length) of the vector using the formula:
\[
|\mathbf{v}| = \sqrt{(x^2 + y^2)}
\]
Plugging in our components \( (6, -2) \):
\[
|\mathbf{v}| = \sqrt{(6^2 + (-2)^2)} = \sqrt{(36 + 4)} = \sqrt{40}
\]
The magnitude can be simplified further:
\[
|\mathbf{v}| = \sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}
\]
Thus, the magnitude of the vector \( \mathbf{v} \) is
\[
\boxed{2\sqrt{10}}
\]