Final answer:
The angle between the force F and displacement d is approximately 68.2°. The projection of F on d is approximately 2.26.
Step-by-step explanation:
To find the angle between force F and displacement d, we can use the dot product formula: θ = cos⁻¹(F · d / |F| * |d|), where · represents the dot product of vectors and | | represents the magnitude of a vector. Using the given values: F = (3î + 4j - 5k) and d = (5î + 4j + 3k), we can calculate:
|F| = sqrt(3² + 4² + (-5)²) = sqrt(50)
|d| = sqrt(5² + 4² + 3²) = sqrt(50)
F · d = 3*5 + 4*4 + (-5)*3 = 15 + 16 - 15 = 16
Now we can substitute the values into the formula: θ = cos⁻¹(16 / (sqrt(50) * sqrt(50))) = cos⁻¹(16 / 50) = cos⁻¹(8 / 25) ≈ 68.2°
To find the projection of F on d, we can use the formula: projection = (F · d) / |d|. Substituting the values: projection = 16 / sqrt(50) ≈ 2.26