Question

A vector is \(20.0 \mathrm{m}\) long and makes an angle of $60.0^{\circ} \mathrm{coun}-\( terclockwise from the \)y\( -axis (on the side of the \)-x$ -axis). What are the \(x\) - and \(y\) -components of this vector?

Short answer

Answer: The x-component of the vector is -10.0 m, and the y-component is 17.32 m.

Step-by-step solution

Determine the reference angle

To find the x and y components, first, we need to find the reference angle from the -x-axis. Since the angle is measured counterclockwise from the y-axis, we know that: Reference angle = \((180^{\circ} - 60^{\circ})= 120^{\circ}\)

Find the x- and y-component using trigonometric functions

Trigonometric functions can be used to find the x- and y-components of the vector. For x-component, \( \textit{x} = \textit{magnitude} \times \textit{cos(reference angle)}\) For y-component, \( \textit{y} = \textit{magnitude} \times \textit{sin(reference angle)}\)

Calculate the x-component

Using the formula for x-component, and given magnitude, and reference angle: x = \(20.0 \ (\textit{m}) \times \textit{cos(120°)}\) x-component = \(20.0 \ (\textit{m}) \times -0.5\) (Note that the cos(120°) = -0.5) x-component = \(-10.0 \mathrm{m}\)

Calculate the y-component

Using the formula for y-component, and given magnitude, and reference angle: y = \(20.0 \ (\textit{m}) \times \textit{sin(120°)}\) y-component = \(20.0 \ (\textit{m}) \times 0.866\) (Note that the sin(120°) = 0.866) y-component = \(17.32 \mathrm{m}\)

Report the components

The x-component of the vector is -10.0 m and the y-component is 17.32 m.