Final Answer:
The magnitude of vector Ā is approximately 54.2 meters, and the angle between its direction and the positive x-axis is about 124.5°.
Thus option a is correct.
Explanation:
To find the magnitude of vector Ā, we use the Pythagorean theorem. Given the x-component (-29.0 m) and y-component (+42.4 m), we square each component, sum the squares, and take the square root of the sum:
Magnitude of Ā = √((-29.0 m)² + (42.4 m)²)
Magnitude of Ā = √(841 + 1797.76)
Magnitude of Ā ≈ √2638.76
Magnitude of Ā ≈ 54.2 m
Now, to determine the angle θ between vector Ā and the positive x-axis, we use trigonometry. The tangent of θ can be found using the ratio of the y-component to the x-component:
tan(θ) = (y-component)/(x-component)
tan(θ) = 42.4 m / (-29.0 m)
θ ≈ tan⁻¹(42.4 m / (-29.0 m))
θ ≈ 124.5°
Therefore, the magnitude of vector Ā is approximately 54.2 meters, and the angle between its direction and the positive x-axis is about 124.5°. This angle is measured counterclockwise from the positive x-axis to the vector's direction in the second quadrant. The magnitude is calculated using the Pythagorean theorem, while the angle is determined using inverse trigonometric functions based on the ratio of the vector's components.
Therefore option a is correct.