Question

A displacement vector $\overrightarrow{\mathbf{r}}$in the xy plane is long and directed at angle $\mathit{\theta }\mathbf{=}{\mathbf{30}}^{\mathbf{°}}$in Fig.3-26 . Determine (a) the x component and (b) the y component of the vector1.

Short answer

The x component of $\overrightarrow{r}$is 1.5 m

The y component of $\overrightarrow{r}$is 7.5 m

Step-by-step solution

Step 1: Calculation for x component of vector r→

Each component of a vector represents a split of the vector. The components of a vector could be computed if the vector is split along each of the axes. Here trigonometric functions and the given magnitude of vectors are used to find components of the vector.

Given the displacement vector $\overrightarrow{r}$is 15 m and the direction of vector is $\theta ={30}^{°}$.

The x and y components of the vector a are given by the following equations.

$$\begin{gathered} r_x=r\cos \theta \left(1\right) \\ {r}_y=r\sin \theta \left(2\right) \end{gathered}$$

Substituting the values of $\overrightarrow{r}$and $\theta$in equation (i), the x component is calculated as,

${\mathrm{r}}_{\mathrm{x}}=15\mathrm{m}\times \cos \left({30}^{°}\right)=15\mathrm{m}.$

Step 2: Calculation for y component of vector r→

Substituting the values of $\overrightarrow{r}$and $\theta$in equation (ii), the x component is calculated as,

${\mathrm{r}}_{\mathrm{y}}=15\mathrm{m}\times \sin \left({30}^{°}\right)=7.5\mathrm{m}$