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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 v \mathbf{v} has an initial point at (−7,4) (-7,4) and a terminal point at (−8,4) (-8,4) , find the exact value of ∥v∥ \|\mathbf{v}\| .\newlineAnswer:

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Full solution

Q. Given the vector v \mathbf{v} has an initial point at (−7,4) (-7,4) and a terminal point at (−8,4) (-8,4) , find the exact value of ∥v∥ \|\mathbf{v}\| .\newlineAnswer:
  1. Define Magnitude of Vector: To find the magnitude of vector vv, we need to calculate the difference between the terminal and initial points in both the xx and yy directions. The magnitude of a vector vv, denoted as ∣∣v∣∣||v||, is given by the formula ∣∣v∣∣=(Δx)2+(Δy)2||v|| = \sqrt{(\Delta x)^2 + (\Delta y)^2}, where Δx\Delta x is the change in the xx-coordinate and Δy\Delta y is the change in the yy-coordinate.
  2. Calculate Δx\Delta x: Calculate Δx\Delta x, which is the change in the x-coordinate. Δx=xterminal−xinitial=−8−(−7)=−8+7=−1\Delta x = x_{\text{terminal}} - x_{\text{initial}} = -8 - (-7) = -8 + 7 = -1.
  3. Calculate Δy\Delta y: Calculate Δy\Delta y, which is the change in the y-coordinate. Δy=yterminal−yinitial=4−4=0\Delta y = y_{\text{terminal}} - y_{\text{initial}} = 4 - 4 = 0.
  4. Calculate Magnitude: Now, we can calculate the magnitude of vector vv using the formula ∣∣v∣∣=(Δx)2+(Δy)2||v|| = \sqrt{(\Delta x)^2 + (\Delta y)^2}. Substituting the values we found, ∣∣v∣∣=(−1)2+(0)2=1+0=1||v|| = \sqrt{(-1)^2 + (0)^2} = \sqrt{1 + 0} = \sqrt{1}.
  5. Final Result: The square root of 11 is 11. Therefore, the magnitude of vector vv is 11.

More problems from Transformations of absolute value functions: translations and reflections

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