Question

What are (a) the x component and (b) the y component of a vector$\overrightarrow{\mathbf{a}}$in the xyplane if its direction iscounterclockwise from the positive direction ${\mathbf{250}}^{\mathbf{°}}$of the x axis and its magnitude is 7.3 m?

Short answer

The x component of $\overrightarrow{a}$is -2.50m

The yx component of $\overrightarrow{a}$is -6.9 m

Step-by-step solution

Step 1: Calculation for x component of vector a→

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 magnitude of vector$\overrightarrow{a}$is 7.3 m and the direction of vector is$\theta ={250}^{°}$counter clockwise from the positive direction of the X axis.

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

$$\begin{gathered} {\mathrm{a}}_{\mathrm{x}}=\mathrm{acos\theta }...\left(1\right) \\ {\mathrm{a}}_{\mathrm{y}}=\mathrm{a}\mathrm{sin\theta }..\left(2\right) \end{gathered}$$

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

${\mathrm{a}}_{\mathrm{x}}=7.3\mathrm{m}\times \cos \left({250}^{°}\right)=-2.50\mathrm{m}$.

Step 2: Calculation for y component of vector a→

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

${\mathrm{a}}_{\mathrm{y}}=7.3\mathrm{m}\times \sin \left({250}^{°}\right)=-6.9\mathrm{m}$