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Given the vector v has an initial point at (8,8) and a terminal point at (8,2), find the exact value of ||v||. Answer:

Given the vector v \mathbf{v} has an initial point at (8,8) (8,8) and a terminal point at (8,2) (8,2) , find the exact value of ∥v∥ \|\mathbf{v}\| .\newlineAnswer:

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Transformations of absolute value functions: translations and reflectionsright chevron icon

Full solution

Q. Given the vector v \mathbf{v} has an initial point at (8,8) (8,8) and a terminal point at (8,2) (8,2) , find the exact value of ∥v∥ \|\mathbf{v}\| .\newlineAnswer:
  1. Calculate Differences and Square: To find the magnitude of vector vv, we need to calculate the difference in the xx-coordinates and the yy-coordinates between the terminal point and the initial point. The magnitude of vector vv, denoted as ∣∣v∣∣||v||, is given by the formula:\newline∣∣v∣∣=((x2−x1)2+(y2−y1)2)||v|| = \sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}\newlinewhere (x1,y1)(x_1, y_1) is the initial point and (x2,y2)(x_2, y_2) is the terminal point.
  2. Substitute Given Points: Substitute the given points into the formula. The initial point is (8,8)(8,8) and the terminal point is (8,2)(8,2). Therefore, x1=8x_1 = 8, y1=8y_1 = 8, x2=8x_2 = 8, and y2=2y_2 = 2.\newline∣∣v∣∣=((8−8)2+(2−8)2)||v|| = \sqrt{((8 - 8)^2 + (2 - 8)^2)}
  3. Simplify Squares and Sum: Calculate the differences and square them.\newline∣∣v∣∣=(0)2+(−6)2||v|| = \sqrt{(0)^2 + (-6)^2}
  4. Calculate Square Root: Simplify the squares and sum them up.\newline∣∣v∣∣=0+36||v|| = \sqrt{0 + 36}
  5. Calculate Square Root: Simplify the squares and sum them up.\newline∣∣v∣∣=(0+36)||v|| = \sqrt{(0 + 36)}Calculate the square root to find the magnitude.\newline∣∣v∣∣=36||v|| = \sqrt{36}\newline∣∣v∣∣=6||v|| = 6

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