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 = 52°, |v| = 63
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 \( |\mathbf{v}| = 63 \) and \( \alpha = 52^\circ \), we calculate \( v_x = 63 \cos(52^\circ) \).
Using a calculator, \( \cos(52^\circ) \approx 0.6157 \), so \( v_x \approx 63 \times 0.6157 \approx 38.79 \approx 39 \) (rounded to nearest integer).

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 \( |\mathbf{v}| = 63 \) and \( \alpha = 52^\circ \), we calculate \( v_y = 63 \sin(52^\circ) \).
Using a calculator, \( \sin(52^\circ) \approx 0.7880 \), so \( v_y \approx 63 \times 0.7880 \approx 49.64 \approx 50 \) (rounded to nearest integer).

Answer:

The magnitude of the horizontal component of \( \mathbf{v} \) is \( \boldsymbol{39} \).
The magnitude of the vertical component of \( \mathbf{v} \) is \( \boldsymbol{50} \).