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Given the vector
v has an initial point at
(-1,3) and a terminal point at
(-2,4), find the exact value of
||v||.
Answer:

Given the vector $\mathbf{v}$ has an initial point at $(-1,3)$ and a terminal point at $(-2,4)$ , 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

(-1,3)

and a terminal point at

(-2,4)

, find the exact value of

\|\mathbf{v}\|

.

\newline

Answer:

Use Distance Formula: To find the magnitude ( $\|\mathbf{v}\|$ ) of the vector $\mathbf{v}$ , we need to use the distance formula, which is derived from the Pythagorean theorem. The distance formula for a vector with initial point $(x_1, y_1)$ and terminal point $(x_2, y_2)$ is: $\newline$ $\|\mathbf{v}\| = \sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}$
Substitute Given Points: Substitute the given points into the distance formula: $\newline$ Initial point: $(-1, 3)$ $\newline$ Terminal point: $(-2, 4)$ $\newline$ $||v|| = \sqrt{((-2 - (-1))^2 + (4 - 3)^2)}$
Simplify Expression: Simplify the expression inside the square root: $\newline$ ||v|| = \sqrt{((-2 + 1)^2 + (4 - 3)^2)}\(\newline||v|| = \sqrt{((-1)^2 + (1)^2)}\)
Calculate Squares: Calculate the squares and sum them up: $\newline$ $||v|| = \sqrt{1 + 1}$
Take Square Root: Add the values inside the square root and then take the square root: $||v|| = \sqrt{2}$

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