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Given the vector v has an initial point at (3,-4) and a terminal point at (6,-1), find the exact value of ||v||. Answer:

Given the vector v \mathbf{v} has an initial point at (3,−4) (3,-4) and a terminal point at (6,−1) (6,-1) , 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 (3,−4) (3,-4) and a terminal point at (6,−1) (6,-1) , 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. Then we will use the Pythagorean theorem to find the magnitude.\newlineCalculation:\newlineΔx=xterminal−xinitial=6−3=3\Delta x = x_{\text{terminal}} - x_{\text{initial}} = 6 - 3 = 3\newlineΔy=yterminal−yinitial=−1−(−4)=3\Delta y = y_{\text{terminal}} - y_{\text{initial}} = -1 - (-4) = 3
  2. Find Magnitude: Now that we have the components of the vector (Δx,Δy)=(3,3)(\Delta x, \Delta y) = (3, 3), we can calculate the magnitude of the vector using the formula ∣∣v∣∣=Δx2+Δy2||v|| = \sqrt{\Delta x^2 + \Delta y^2}.\newlineCalculation:\newline∣∣v∣∣=32+32=9+9=18||v|| = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18}
  3. Simplify Result: We can simplify 18\sqrt{18} by factoring out the square of the largest perfect square factor of 1818, which is 99.\newlineCalculation:\newline18=9×2=9×2=32\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}

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