To determine the angle that a vector [tex]\( R \)[/tex] makes with the horizontal given the ratio of its components [tex]\( \frac{R_x}{R_y} = 2 \)[/tex], we follow these steps:
1. Understand the ratio:
The given ratio is [tex]\(\frac{R_x}{R_y} = 2\)[/tex], which implies that [tex]\( R_x = 2 R_y \)[/tex].
2. Express the angle [tex]\( \theta \)[/tex]:
The angle [tex]\( \theta \)[/tex] that the vector [tex]\( R \)[/tex] makes with the horizontal (usually the x-axis) can be found using the arctangent function. This angle can be expressed as:
[tex]\[
\theta = \arctan\left(\frac{R_y}{R_x}\right)
\][/tex]
3. Substitute the given ratio into the arctangent function:
Since [tex]\( R_x = 2 R_y \)[/tex], we substitute this into the equation:
[tex]\[
\theta = \arctan\left(\frac{R_y}{2 R_y}\right) = \arctan\left(\frac{1}{2}\right)
\][/tex]
4. Calculate the angle:
Evaluating [tex]\(\arctan\left(\frac{1}{2}\right)\)[/tex] results in:
[tex]\[
\theta \approx 26.56^\circ
\][/tex]
Therefore, the angle that the resultant vector makes with the horizontal is approximately [tex]\( 26.56^\circ \)[/tex].
Given the options:
- A. [tex]\( 60.02^\circ \)[/tex]
- B. [tex]\( 52.60^\circ \)[/tex]
- C. [tex]\( 26.56^\circ \)[/tex]
- D. [tex]\( 69.59^\circ \)[/tex]
- E. [tex]\( 72.35^\circ \)[/tex]
The correct answer is:
C. [tex]\( 26.56^\circ \)[/tex]
