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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 v \mathbf{v} has an initial point at (7,6) (7,6) and a terminal point at (1,0) (1,0) , find the exact value of ∥v∥ \|\mathbf{v}\| .\newlineAnswer:

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Q. Given the vector v \mathbf{v} has an initial point at (7,6) (7,6) and a terminal point at (1,0) (1,0) , find the exact value of ∥v∥ \|\mathbf{v}\| .\newlineAnswer:
  1. Calculate Components: To find the magnitude of vector vv, we need to calculate the difference in the xx-coordinates and the yy-coordinates of the initial and terminal points to get the components of the vector.\newlineCalculation:\newlineΔx=xterminal−xinitial=1−7=−6\Delta x = x_{\text{terminal}} - x_{\text{initial}} = 1 - 7 = -6\newlineΔy=yterminal−yinitial=0−6=−6\Delta y = y_{\text{terminal}} - y_{\text{initial}} = 0 - 6 = -6
  2. Use Pythagorean Theorem: Now, we use the Pythagorean theorem to find the magnitude of vector vv, which is the square root of the sum of the squares of its components.\newlineCalculation:\newline∣∣v∣∣=Δx2+Δy2=(−6)2+(−6)2=36+36=72||v|| = \sqrt{\Delta x^2 + \Delta y^2} = \sqrt{(-6)^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72}
  3. Simplify Result: We can simplify 72\sqrt{72} by factoring out the largest perfect square, which is 3636, and then taking the square root of that factor outside the radical.\newlineCalculation:\newline72=(36×2)=36×2=62\sqrt{72} = \sqrt{(36 \times 2)} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}

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