How can I find the angle between the vectors u = (8, 7) and v = (9, 7) to the nearest tenth of a degree?

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To determine the angle between the two vectors u and v, you can use the dot product formula along with the magnitudes of the vectors. The dot product formula is given by:

u · v = |u| |v| cos(θ)

Where:

  • u · v is the dot product of the vectors u and v.
  • |u| and |v| are the magnitudes of the vectors u and v.
  • θ is the angle between the two vectors.

First, let’s calculate the dot product of the vectors u and v: 

u · v = (8 * 9) + (7 * 7) = 72 + 49 = 121

Next, we need to find the magnitudes of the two vectors:

|u| = √(82 + 72) = √(64 + 49) = √(113) ≈ 10.630
|v| = √(92 + 72) = √(81 + 49) = √(130) ≈ 11.401

Now, substituting the dot product and magnitudes into the initial formula:

121 = (10.630 * 11.401) cos(θ)

Calculating the right side:

10.630 * 11.401 ≈ 121.073

Now we can find cos(θ):

cos(θ) = 121 / 121.073 ≈ 0.999

To find the angle θ, take the inverse cosine:

θ ≈ cos-1(0.999)

Calculating this gives:

θ ≈ 2.3 degrees

Thus, the angle between the vectors u = (8, 7) and v = (9, 7) is approximately 2.3 degrees when rounded to the nearest tenth.

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