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Given the vector
v has an initial point at
(-7,4) and a terminal point at
(-8,4), find the exact value of
||v||.
Answer:

Given the vector $\mathbf{v}$ has an initial point at $(-7,4)$ and a terminal point at $(-8,4)$ , find the exact value of $\|\mathbf{v}\|$ . $\newline$ Answer:

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Transformations of absolute value functions: translations and reflections

Full solution

Q. Given the vector

\mathbf{v}

has an initial point at

(-7,4)

and a terminal point at

(-8,4)

, find the exact value of

\|\mathbf{v}\|

.

\newline

Answer:

Define Magnitude of Vector: To find the magnitude of vector $v$ , we need to calculate the difference between the terminal and initial points in both the $x$ and $y$ directions. The magnitude of a vector $v$ , denoted as $||v||$ , is given by the formula $||v|| = \sqrt{(\Delta x)^2 + (\Delta y)^2}$ , where $\Delta x$ is the change in the $x$ -coordinate and $\Delta y$ is the change in the $y$ -coordinate.
Calculate $\Delta x$ : Calculate $\Delta x$ , which is the change in the x-coordinate. $\Delta x = x_{\text{terminal}} - x_{\text{initial}} = -8 - (-7) = -8 + 7 = -1$ .
Calculate $\Delta y$ : Calculate $\Delta y$ , which is the change in the y-coordinate. $\Delta y = y_{\text{terminal}} - y_{\text{initial}} = 4 - 4 = 0$ .
Calculate Magnitude: Now, we can calculate the magnitude of vector $v$ using the formula $||v|| = \sqrt{(\Delta x)^2 + (\Delta y)^2}$ . Substituting the values we found, $||v|| = \sqrt{(-1)^2 + (0)^2} = \sqrt{1 + 0} = \sqrt{1}$ .
Final Result: The square root of $1$ is $1$ . Therefore, the magnitude of vector $v$ is $1$ .

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