Final answer:
To find the magnitude and direction of vector a, we use the dot product property and solve for the unknowns. The magnitude of vector a is 10.91 m and the direction is 85.0°.
Step-by-step explanation:
To find the magnitude and direction of vector a, we can use the dot product property. Since a · b = 27.0 m², we can write a · b = |a| |b| cos(θ), where |a| and |b| are the magnitudes of vectors a and b, and θ is the angle between them.
Given that b = 4.95 m and θ = 60.0°, we can solve for |a|:
27.0 m² = |a| * 4.95 m * cos(60.0°)
27.0 m² = 4.95 m |a| * 0.5
|a| = 27.0 m² / (4.95 m * 0.5)
|a| = 27.0 m² / 2.475 m
|a| = 10.91 m
To find the direction of vector a, we can use the fact that the direction angle of c is larger than that of a by 25.0°. Since the direction angle of c is not given, we can calculate it using the dot product property b · c = |b| |c| cos(θ), where θ is the angle between b and c. Given that b · c = 31.5 m², we can write:
31.5 m² = 4.95 m |c| cos(θ)
31.5 m² = 4.95 m |c| * 0.5
|c| = 31.5 m² / (4.95 m * 0.5)
|c| = 31.5 m² / 2.475 m
|c| = 12.73 m
Since a and c have equal magnitudes, |a| = |c| = 10.91 m. Therefore, the magnitude of vector a is 10.91 m.
The direction of vector a is 60.0° + 25.0° = 85.0°