The answer is 2.82 radians.
To find the angle between two vectors, u and v, we can use the dot product formula:
u · v = |u| |v| cos(θ),
where u · v represents the dot product of u and v, |u| and |v| represent the magnitudes of u and v, and θ represents the angle between the vectors.
Let's calculate the dot product:
u · v = (8)(5) + (-7)(9)
= 40 - 63
= -23.
Next, let's calculate the magnitudes of u and v:
|u| = sqrt((8^2) + (-7^2))
= sqrt(64 + 49)
= sqrt(113).
|v| = sqrt((5^2) + (9^2))
= sqrt(25 + 81)
= sqrt(106).
Now, let's substitute the values into the dot product formula and solve for θ:
-23 = sqrt(113) sqrt(106) cos(θ).
Dividing both sides by sqrt(113) sqrt(106), we have:
cos(θ) = -23 / (sqrt(113) sqrt(106)).
Now we can find the angle θ by taking the inverse cosine (arccos) of the right-hand side:
θ = arccos(-23 / (sqrt(113) sqrt(106))).
Using a calculator or a trigonometric table, we can find the approximate value of θ to two decimal places:
θ ≈ 2.82 radians.
Therefore, the angle between the vectors u = 8i - 7j and v = 5i + 9j is approximately 2.82 radians.