Select the correct answer. A vector [tex]$R$[/tex] is resolved into its components, [tex]$R_x$[/tex] and [tex]$R_y$[/tex]. If the ratio of [tex]$\frac{R_x}{R_y}$[/tex] is 2, what is the angle that the resultant makes with the horizontal?
A. [tex]$60.02^{\circ}$[/tex]
B. [tex]$5360^{\circ}$[/tex]
C. [tex]$26.56^{\circ}$[/tex]
D. [tex]$69.59^*$[/tex]
E. [tex]$7235^{\circ}$[/tex]
Answer (2)
The problem provides the ratio of the x and y components of a vector.
We find the tangent of the angle with the horizontal using the inverse of the given ratio: tan ( θ ) = R x R y = 2 1 .
Calculate the angle by taking the inverse tangent: θ = arctan ( 2 1 ) .
The angle that the resultant makes with the horizontal is approximately: 26.5 6 ∘ .
Explanation
Problem Analysis We are given that a vector R is resolved into its components R x and R y , and the ratio R y R x = 2 . We need to find the angle θ that the resultant vector R makes with the horizontal.
Finding the Tangent of the Angle The tangent of the angle θ that the resultant vector R makes with the horizontal (x-axis) is given by: tan ( θ ) = R x R y Since R y R x = 2 , we have: R x R y = 2 1 Therefore, tan ( θ ) = 2 1
Calculating the Angle To find the angle θ , we take the inverse tangent (arctan) of 2 1 :
θ = arctan ( 2 1 ) Calculating the value of arctan ( 2 1 ) in degrees, we get: θ ≈ 26.5 6 ∘
Examples
Understanding vector components and angles is crucial in many real-world applications. For example, when analyzing the trajectory of a projectile, like a ball thrown in the air, we need to resolve the initial velocity vector into horizontal and vertical components. The angle at which the ball is thrown determines the range and height it will achieve. Similarly, in navigation, pilots and sailors use vector addition and angles to calculate the course and speed of their aircraft or vessel, taking into account wind or current.
The angle that the resultant vector R makes with the horizontal is approximately 26.5 6 ∘ . This is derived from the ratio of its components, where R x is twice R y . Therefore, the correct multiple-choice answer is C. 26.5 6 ∘ .
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