Vector v has the given direction angle and magnitude. Find the horizontal and vertical components.
θ = 27° 30' |v| = 15.4

Verified step by step guidance

1. Convert the angle from degrees and minutes to decimal degrees. Since 1 minute is 1/60 of a degree, convert 30 minutes to degrees: 30' = 30/60 = 0.5 degrees. Therefore, \( \theta = 27.5^\circ \).
2. Use the magnitude \( |v| = 15.4 \) and the angle \( \theta = 27.5^\circ \) to find the horizontal component \( v_x \) using the formula \( v_x = |v| \cdot \cos(\theta) \).
3. Calculate the vertical component \( v_y \) using the formula \( v_y = |v| \cdot \sin(\theta) \).
4. Substitute the values into the formulas: \( v_x = 15.4 \cdot \cos(27.5^\circ) \) and \( v_y = 15.4 \cdot \sin(27.5^\circ) \).
5. Evaluate the trigonometric functions and multiply by the magnitude to find the horizontal and vertical components.

Key Concepts

Direction Angle

The direction angle of a vector is the angle formed between the positive x-axis and the line representing the vector. It is typically measured in degrees or radians and indicates the orientation of the vector in a two-dimensional plane. In this case, the angle θ = 27° 30' specifies how the vector is inclined relative to the horizontal axis.

Magnitude of a Vector

The magnitude of a vector represents its length or size, denoted as |v|. It is a scalar quantity that indicates how far the vector extends from the origin to its endpoint. In this problem, the magnitude is given as 15.4, which is essential for calculating the horizontal and vertical components of the vector.

Horizontal and Vertical Components

The horizontal (x) and vertical (y) components of a vector can be found using trigonometric functions. Specifically, the horizontal component is calculated as |v| * cos(θ) and the vertical component as |v| * sin(θ). These components allow us to express the vector in terms of its influence along the x and y axes, facilitating further analysis and calculations.