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 direction angle from degrees and minutes to decimal degrees. Since 1 minute is \( \frac{1}{60} \) of a degree, calculate \( 27^\circ 30' = 27 + \frac{30}{60} = 27.5^\circ \).
2. Recall that the horizontal component (\( v_x \)) of a vector is found by multiplying the magnitude by the cosine of the direction angle: \( v_x = |v| \cdot \cos(\theta) \).
3. Similarly, the vertical component (\( v_y \)) is found by multiplying the magnitude by the sine of the direction angle: \( v_y = |v| \cdot \sin(\theta) \).
4. Substitute the given magnitude \( |v| = 15.4 \) and the converted angle \( \theta = 27.5^\circ \) into the formulas: \( v_x = 15.4 \cdot \cos(27.5^\circ) \) and \( v_y = 15.4 \cdot \sin(27.5^\circ) \).
5. Use a calculator set to degree mode to evaluate the cosine and sine values, then multiply to find the horizontal and vertical components of the vector.

Key Concepts

Direction Angle in Vectors

The direction angle of a vector is the angle it makes with the positive x-axis, measured counterclockwise. It determines the vector's orientation in the plane and is essential for decomposing the vector into components along the horizontal and vertical axes.

Vector Magnitude

The magnitude of a vector represents its length or size. It is a scalar quantity that, combined with the direction angle, fully describes the vector. Knowing the magnitude allows calculation of the vector's components using trigonometric functions.

Resolving Vectors into Components

A vector can be broken down into horizontal (x) and vertical (y) components using trigonometry. The horizontal component is found by multiplying the magnitude by the cosine of the direction angle, and the vertical component by the sine of the angle, enabling easier analysis and computation.