Find the angle between the given vectors to the nearest tenth of a degree. u = <-5, 8>, v = <-4, 8>

2.7°
-7.3°
5.4°
15.4°

9514 1404 393

  (c) 5.4°

Step-by-step explanation:

We can simply determine the difference between the angles of the vectors:

  arctan(8/-5) -arctan(8/-4) = 122.005° -116.565° = 5.440°

The angle between the vectors is about 5.4°.

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Alternate solutions

1) We could also determine the angle of their ratio.

  u/v = (-5 +8i)/(-4 +8i) = (-5 +8i)(-4 -8i)/(4² +8²) = (84 +8i)/80 = 1.05 +0.1i

  angle = arctan(0.1/1.05) = 5.440°

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2) Tan(a-b) = (tan(a)+tan(b))/(1 -tan(a)(tan(b))

  tan(∠u -∠v) = ((tan(∠u) -tan(∠v))/(1 +tan(∠u)tan(∠v))

  = (-1.6 -(-2))/(1 +(-1.6)(-2)) = 0.4/4.2

  ∠u -∠v = arctan(2/21) = 5.440°