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Given the vector
v has an initial point at
(7,6) and a terminal point at
(1,0), find the exact value of
||v||.
Answer:

Given the vector $\mathbf{v}$ has an initial point at $(7,6)$ and a terminal point at $(1,0)$ , 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

(7,6)

and a terminal point at

(1,0)

, find the exact value of

\|\mathbf{v}\|

.

\newline

Answer:

Calculate Components: To find the magnitude of vector $v$ , we need to calculate the difference in the $x$ -coordinates and the $y$ -coordinates of the initial and terminal points to get the components of the vector. $\newline$ Calculation: $\newline$ $\Delta x = x_{\text{terminal}} - x_{\text{initial}} = 1 - 7 = -6$ $\newline$ $\Delta y = y_{\text{terminal}} - y_{\text{initial}} = 0 - 6 = -6$
Use Pythagorean Theorem: Now, we use the Pythagorean theorem to find the magnitude of vector $v$ , which is the square root of the sum of the squares of its components. $\newline$ Calculation: $\newline$ $||v|| = \sqrt{\Delta x^2 + \Delta y^2} = \sqrt{(-6)^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72}$
Simplify Result: We can simplify $\sqrt{72}$ by factoring out the largest perfect square, which is $36$ , and then taking the square root of that factor outside the radical. $\newline$ Calculation: $\newline$ $\sqrt{72} = \sqrt{(36 \times 2)} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$

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