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In a triangle where b = 4.95 m at 60.0°, c and a have equal magnitudes. The direction angle of c is larger than that of a by 25.0°. If a · b = 27.0 m² and b · c = 31.5 m², find the magnitude (in m) and direction (in degrees) of a.

a) Magnitude: 16.2, Direction: 35.0°
b) Magnitude: 18.7, Direction: 55.0°
c) Magnitude: 15.0, Direction: 80.0°
d) Magnitude: 20.5, Direction: 110.0°

1 Answer

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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°

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