Vector A has a magnitude of 5.00 units, vector B has a magnitude of 9.00 units, and the dot product A·B has a value of 42. What is the angle between the directions of A and B?

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To find the angle between two vectors A and B, you can use the dot product formula:

A · B = |A| |B| cos(θ)

Where A · B is the dot product of A and B, |A| and |B| are the magnitudes of A and B respectively, and θ is the angle between the two vectors.

In this case, A · B = 42, |A| = 5.00, and |B| = 9.00. We need to find θ.

By rearranging the formula and solving for θ, we have:

cos(θ) = (A · B) / (|A| |B|)
cos(θ) = 42 / (5.00 * 9.00)
cos(θ) = 42 / 45.00
cos(θ) ≈ 0.933

Now we need to find the inverse cosine (aka arccos or cos^(-1)) of 0.933 to find the angle θ:

θ ≈ cos^(-1)(0.933) ≈ 21.8 degrees

Therefore, the angle between the directions of vectors A and B is approximately 21.8 degrees.