Question

Find the magnitude of vector v that has an initial point of left parenthesis 1 comma 3 right parenthesis and a terminal point of left parenthesis 7 comma 1 right parenthesis.
(1 point)
Responses

double vertical line bold v double vertical line equals 2 start root 10 end root
Image with alt text: double vertical line bold v double vertical line equals 2 start root 10 end root

double vertical line bold v double vertical line equals 2 start root 14 end root
Image with alt text: double vertical line bold v double vertical line equals 2 start root 14 end root

double vertical line bold v double vertical line equals 4 start root 2 end root
Image with alt text: double vertical line bold v double vertical line equals 4 start root 2 end root

double vertical line bold v double vertical line equals 4 start root 5 end root

Answer 1

To find the magnitude of vector \(\mathbf{v}\), we first need to determine the components of the vector using its initial and terminal points.

The initial point is \( (1, 3) \) and the terminal point is \( (7, 1) \).

The components of vector \(\mathbf{v}\) can be calculated as follows:

\[
\mathbf{v} = \langle x_2 - x_1, y_2 - y_1 \rangle = \langle 7 - 1, 1 - 3 \rangle = \langle 6, -2 \rangle
\]

Next, we find the magnitude of vector \(\mathbf{v}\), which is calculated using the formula:

\[
\|\mathbf{v}\| = \sqrt{(x^2 + y^2)} = \sqrt{(6)^2 + (-2)^2}
\]

Calculating this gives us:

\[
\|\mathbf{v}\| = \sqrt{36 + 4} = \sqrt{40}
\]

We can simplify \(\sqrt{40}\):

\[
\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}
\]

Therefore, the magnitude of vector \(\mathbf{v}\) is:

\[
\|\mathbf{v}\| = 2\sqrt{10}
\]

So the correct response is:

\[
\boxed{2\sqrt{10}}
\]