Question

A vector \(\overrightarrow{\mathbf{A}}\) has a magnitude of \(22.2 \mathrm{cm}\) and makes an angle of \(130.0^{\circ}\) with the positive \(x\) -axis. What are the \(x\) - and \(y\) -components of this vector?

Short answer

Answer: The x-component of the vector is approximately -15.82 cm, and the y-component is approximately 17.71 cm.

Step-by-step solution

Identify given information

We are given the magnitude of vector \(\overrightarrow{\mathbf{A}}\), which is \(22.2 \mathrm{cm}\), and the angle it makes with the positive \(x\)-axis, which is \(130.0^{\circ}\). We will use this information to find the \(x\)- and \(y\)-components of the vector.

Calculate the x-component of the vector

To find the \(x\)-component of the vector, we will use the cosine function, as it is related to the angle and adjacent side. The formula is: $$A_x = A \cos(\theta)$$ where \(A_x\) is the \(x\)-component, \(A\) is the magnitude of the vector, and \(\theta\) is the angle. Plugging in the given values, we get: $$A_x = 22.2 \mathrm{cm} \cos(130.0^{\circ})$$ Now, calculate the value of \(A_x\): $$A_x \approx -15.82 \mathrm{cm}$$

Calculate the y-component of the vector

Similarly, to find the \(y\)-component of the vector, we will use the sine function, as it is related to the angle and opposite side. The formula is: $$A_y = A \sin(\theta)$$ where \(A_y\) is the \(y\)-component. Plugging in the given values, we get: $$A_y = 22.2 \mathrm{cm} \sin(130.0^{\circ})$$ Now, calculate the value of \(A_y\): $$A_y \approx 17.71 \mathrm{cm}$$

State the x- and y-components of the vector

After performing the calculations, we found that the \(x\)- and \(y\)-components of the vector \(\overrightarrow{\mathbf{A}}\) are: $$A_x \approx -15.82 \mathrm{cm}$$ $$A_y \approx 17.71 \mathrm{cm}$$