How do I find the angle between the vectors u = (2, 4) and v = (3, 8) to the nearest tenth of a degree?

To find the angle between two vectors, you can use the dot product formula. The dot product of two vectors u and v can be calculated as:

u ⋅ v = |u| |v| cos(θ)

Where:

• u ⋅ v is the dot product of the vectors.
• |u| and |v| are the magnitudes (lengths) of the vectors.
• θ is the angle between the vectors.

First, let’s calculate the dot product of the vectors u = (2, 4) and v = (3, 8):

u ⋅ v = (2 * 3) + (4 * 8) = 6 + 32 = 38

Next, we find the magnitudes of the vectors:

• |u| = √((2²) + (4²)) = √(4 + 16) = √(20) ≈ 4.47
• |v| = √((3²) + (8²)) = √(9 + 64) = √(73) ≈ 8.54

Now we can use the dot product and the magnitudes to find the cosine of the angle:

cos(θ) = (u ⋅ v) / (|u| * |v|)

Substituting the values:

cos(θ) = 38 / (4.47 * 8.54) ≈ 38 / 38.2 ≈ 0.994

Now, we can find the angle θ by taking the inverse cosine:

θ = cos-1(0.994)

Using a calculator, we find:

θ ≈ 7.1 degrees

Therefore, the angle between the vectors u and v is approximately 7.1 degrees when rounded to the nearest tenth.