Final answer:
To find the angle between two vectors, u and v, use the dot product formula u · v = |u| * |v| * cos(theta). Solve for theta to find the angle between u and v. For the given vectors u = 3i + 4j and v = -7i + 5j, the angle is approximately 127.44 degrees.
Step-by-step explanation:
To find the angle between two vectors, u and v, we can use the dot product formula:
u · v = |u| * |v| * cos(theta)
where |u| and |v| are the magnitudes of the vectors and theta is the angle between them. In this case, u = 3i + 4j and v = -7i + 5j.
The dot product of u and v is (3 * -7) + (4 * 5) = -21 + 20 = -1.
The magnitudes of u and v are |u| = sqrt(3² + 4²) = 5 and |v| = sqrt((-7)² + 5²) = sqrt(74).
Substituting these values into the formula, we get:
-1 = 5 * sqrt(74) * cos(theta)
To solve for theta, divide both sides by (5 * sqrt(74)) and take the inverse cosine of both sides:
cos(theta) = -1 / (5 * sqrt(74))
theta = acos(-1 / (5 * sqrt(74)))
Using a calculator, theta ≈ 127.44 degrees.