Question

find the magnitudes of the horizontal and vertical components of v, if α is the direction angle of v from the horizontal. α = 75°10, |v| = 14.1 the horizontal component of v is □. (round to the nearest tenth as needed.) the vertical component of v is □. (round to the nearest tenth as needed.)

Explanation:

Step1: Convert angle to decimal degrees

First, convert \( 75^\circ 10' \) to decimal degrees. Since \( 10'=\frac{10}{60}\approx0.1667^\circ \), so \( 75^\circ 10' = 75 + \frac{10}{60}\approx75.1667^\circ \).

Step2: 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 \( v_x = |\mathbf{v}| \cos(\alpha) \).
Substitute \( |\mathbf{v}| = 14.1 \) and \( \alpha\approx75.1667^\circ \) into the formula:
\( v_x = 14.1\times\cos(75.1667^\circ) \)
Calculate \( \cos(75.1667^\circ)\approx\cos(75.1667)\approx0.256 \) (using calculator)
Then \( v_x = 14.1\times0.256\approx3.6 \) (rounded to nearest tenth)

Step3: 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 \( v_y = |\mathbf{v}| \sin(\alpha) \).
Substitute \( |\mathbf{v}| = 14.1 \) and \( \alpha\approx75.1667^\circ \) into the formula:
\( v_y = 14.1\times\sin(75.1667^\circ) \)
Calculate \( \sin(75.1667^\circ)\approx\sin(75.1667)\approx0.966 \) (using calculator)
Then \( v_y = 14.1\times0.966\approx13.6 \) (rounded to nearest tenth)

Answer:

The horizontal component of \( \mathbf{v} \) is \( \boxed{3.6} \).
The vertical component of \( \mathbf{v} \) is \( \boxed{13.6} \).