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Given the vector
v has an initial point at
(0,-3) and a terminal point at
(1,1), find the exact value of
||v||.
Answer:

Given the vector $\mathbf{v}$ has an initial point at $(0,-3)$ and a terminal point at $(1,1)$ , 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

(0,-3)

and a terminal point at

(1,1)

, find the exact value of

\|\mathbf{v}\|

.

\newline

Answer:

Define Initial and Terminal Points: To find the magnitude of vector $v$ , we need to calculate the difference in the $x$ -coordinates and the difference in the $y$ -coordinates between the terminal point and the initial point. Then, we will use the Pythagorean theorem to find the magnitude. $\newline$ Let's denote the initial point as $(x_1, y_1) = (0, -3)$ and the terminal point as $(x_2, y_2) = (1, 1)$ .
Calculate Differences: Calculate the difference in the x-coordinates ( $\Delta x$ ) and the difference in the y-coordinates ( $\Delta y$ ). $\newline$ $\Delta x = x_2 - x_1 = 1 - 0 = 1$ $\newline$ $\Delta y = y_2 - y_1 = 1 - (-3) = 1 + 3 = 4$
Use Pythagorean Theorem: Now, we use the Pythagorean theorem to find the magnitude of vector $v$ , which is denoted as $||v||$ . $||v|| = \sqrt{(\Delta x^2 + \Delta y^2)} = \sqrt{(1^2 + 4^2)} = \sqrt{(1 + 16)} = \sqrt{17}$

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