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Chapter 1: Problem 76

What angle does \(\vec{A}=\left(A_{x}, A_{y}\right)=(30.0 \mathrm{~m},-50.0 \mathrm{~m})\) make with the positive \(x\) -axis? What angle does it make with the negative \(y\) -axis?

Short Answer

Expert verified
Answer: The angle between vector A and the positive x-axis is \(301^{\circ}\), and the angle between vector A and the negative y-axis is \(149^{\circ}\).

Step by step solution

01

Find the angle between vector A and the positive x-axis

We have vector A with components \(A_x = 30.0 m\) and \(A_y = -50.0 m\). To find the angle between A and the positive x-axis, we use the arctangent function: \(\tan^{-1}\left(\frac{A_y}{A_x}\right)\) Calculate the arctangent of \(\frac{-50.0}{30.0}\), which can be done using a calculator: \(\theta_1 = \tan^{-1}\left(\frac{-50.0}{30.0}\right) = -59.0^{\circ}\)
02

Convert the angle to a positive value (if needed)

If the angle found in Step 1 is negative, add 360 degrees so that the angle represents a positive angle measured from the positive x-axis counterclockwise: \(\theta_1 = -59.0^{\circ} + 360^{\circ} = 301^{\circ}\) So, the angle between vector A and the positive x-axis is \(301^{\circ}\).
03

Find the angle between vector A and the negative y-axis

Now, we need to find the angle between vector A and the negative y-axis. To do this, simply subtract the angle found in Step 1 from 90 degrees: \(\theta_2 = 90^{\circ} - (-59.0^{\circ}) = 149^{\circ}\) The angle between vector A and the negative y-axis is \(149^{\circ}\).

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