Question

What are (a) the x-component and (b) y-component of a vector \[\vec a\] 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 \[\vec a\] 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 \[{\vec a_x}\] represents the x component of vector \[\vec a\] then it can be represented as,
\[{\vec a_x}{\text{ = |}}\vec a|{\text{ }} \times {\text{ cos}}\theta \]
Where, \[{\text{|}}\vec a|\] represents the magnitude of the vector \[\vec a\] and \[\theta \] is the angle between the vector and positive x axis.

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

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

Therefore we have calculated the x and y component of a vector called \[\vec a\].

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 \[\dfrac{\pi }{{180}}\]. 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.