Final Answer:
The dot product of vectors u and v is 20. b) The magnitude of vector u is 5.83, and the magnitude of vector v is 4. c) The cosine of the angle between vectors u and v is approximately 0.86. d) The vectors u and v are not orthogonal (Option a).
Step-by-step explanation:
a) The dot product of two vectors u and v is calculated using the formula u ⋅ v = u₁v₁ + u₂v₂. For vectors u = (3, 5) and v = (4, 0), the dot product is 3 × 4 + 5 × 0 = 12. Therefore, the dot product of vectors u and v is 12 (Option a).
b) The magnitude of a vector is given by the formula |u| = √(u₁² + u₂²). For vector u = (3, 5), the magnitude is √(3² + 5²) = √(9 + 25) = √34, which is approximately 5.83. Similarly, for vector v = (4, 0), the magnitude is √(4² + 0²) = 4.
c) The cosine of the angle θ between two vectors u and v is calculated using the formula cos(θ) = (u ⋅ v) / (|u| ⋅ |v|). Substituting the values, cos(θ) = 12 / (5.83 × 4), which is approximately 0.86.
d) Vectors u and v are orthogonal if their dot product is zero. Since the dot product is 12, they are not orthogonal. The nonzero dot product implies that the vectors have some degree of alignment.
In summary, the dot product, magnitude, cosine of the angle, and orthogonality status of vectors u and v have been calculated. These calculations involve basic vector operations and trigonometric functions, providing a comprehensive understanding of the relationship between the two vectors.