Question

find the magnitudes of the horizontal and vertical components for the vector v, if α is the direction angle of v from the horizontal. α = 31°, ||v|| = 26 the magnitude of the horizontal component of v is (round to the nearest integer as needed.) the magnitude of the vertical component of v is (round to the nearest integer as needed.)

Explanation:

Step1: Find horizontal component

The formula for the horizontal component \( v_x \) of a vector \( \mathbf{v} \) with magnitude \( |\mathbf{v}| \) and direction angle \( \alpha \) is \( v_x = |\mathbf{v}| \cos(\alpha) \).
Given \( |\mathbf{v}| = 26 \) and \( \alpha = 31^\circ \), we calculate:
\( v_x = 26 \cos(31^\circ) \)
Using a calculator, \( \cos(31^\circ) \approx 0.8572 \), so \( v_x \approx 26 \times 0.8572 \approx 22.3872 \). Rounding to the nearest integer, \( v_x \approx 22 \).

Step2: Find vertical component

The formula for the vertical component \( v_y \) of a vector \( \mathbf{v} \) with magnitude \( |\mathbf{v}| \) and direction angle \( \alpha \) is \( v_y = |\mathbf{v}| \sin(\alpha) \).
Given \( |\mathbf{v}| = 26 \) and \( \alpha = 31^\circ \), we calculate:
\( v_y = 26 \sin(31^\circ) \)
Using a calculator, \( \sin(31^\circ) \approx 0.5150 \), so \( v_y \approx 26 \times 0.5150 \approx 13.39 \). Rounding to the nearest integer, \( v_y \approx 13 \).

Answer:

Horizontal component: 22
Vertical component: 13