Question

What are (a) the x-component and (b) y-component of a vector $$\overrightarrow{a}$$ in the xy plane if is direction is $${250}^{\circ }$$ counterclockwise from the positive direction of the x-axis and its magnitude is given as $$7.3\text{ m ?}$$

Answer 1

Hint: The magnitude of vector $$\overrightarrow{a}$$ is given. We have to find its x and y components. For any vector the x component of a vector is basically the product of the magnitude of the vector and the cosine of the angle which it makes with the positive direction of the x-axis. The direction of angle here is counter clockwise which means anti-clockwise.

Complete step by step answer:
Since we know that the x component of a vector is the product of the magnitude of the vector and the cosine of the angle which it makes with the positive direction of the x-axis.

(a) Let us assume $${\overrightarrow{a}}_x$$ represents the x component of vector $$\overrightarrow{a}$$ then it can be represented as,
$${\overrightarrow{a}}_x\text{ = |}\overrightarrow{a}|\times \text{ cos}\theta$$
Where, $$\text{|}\overrightarrow{a}|$$ represents the magnitude of the vector $$\overrightarrow{a}$$ and $$\theta$$ is the angle between the vector and positive x axis.

According to question, $$\text{|}\overrightarrow{a}|\text{ = 7}\text{.3}$$ and $$\theta \text{ = 25}{\text{0}}^{\circ }$$ , therefore on
substituting the values we get the result as,
$${\overrightarrow{a}}_x\text{ = |}\overrightarrow{a}|\times \text{ cos}\theta$$
$$\Rightarrow {\overrightarrow{a}}_x\text{ = 7}\text{.3 }\times \text{ cos }\left(\text{25}{\text{0}}^{\circ }\right)$$
$$\Rightarrow {\overrightarrow{a}}_x\text{ = 7}\text{.3 }\times \left(-0.34\right)$$
$$\Rightarrow {\overrightarrow{a}}_x\text{ = - 2}\text{.482 m}$$

(b) Similarly we can find y component of vector $$\overrightarrow{a}$$ as,
$${\overrightarrow{a}}_y\text{ = |}\overrightarrow{a}|\times \text{ sin}\theta$$
Here we multiply $$\text{sin}\theta$$ with the magnitude of vector $$\overrightarrow{a}$$ and we know that according to question, $$\text{|}\overrightarrow{a}|\text{ = 7}\text{.3}$$ and $$\theta \text{ = 25}{\text{0}}^{\circ }$$ , therefore on substituting the values we get the result as,
$${\overrightarrow{a}}_y\text{ = |}\overrightarrow{a}|\times \text{ sin}\theta$$
$$\Rightarrow {\overrightarrow{a}}_y\text{ = 7}\text{.3 }\times \text{ sin }\left({250}^{\circ }\right)$$
$$\Rightarrow {\overrightarrow{a}}_y\text{ = 7}\text{.3 }\times \left(-0.93\right)$$
$$\therefore {\overrightarrow{a}}_y\text{ = - 6}\text{.789 m}$$

Therefore we have calculated the x and y component of a vector called $$\overrightarrow{a}$$.

Note: The value of sine and cosine of the given angle can be find out with the help of trigonometric tables and we can also convert them in terms of $$\pi$$ by multiplying the angle with $${\displaystyle \frac{\pi }{180}}$$. We can also round off these values for easy calculations.The magnitude of a vector is basically the length of a vector in a particular direction, therefore it is measured in meters.