Question

How will you find the length and direction of the vector \[2 - 4i\]?

Answer 1

Hint:The square root of the sum of the squares of the horizontal and vertical components gives the length of vector and an angle \[\theta \] in the standard position of the terminal side though the origin pointing with coordinates determines the direction of vector. This angle \[\theta \] can be found by solving the trigonometric equation.

Formula Used:
In order to find the length of the vector we will use the distance formula or Pythagorean Theorem which is \[\sqrt {{x^2} + {y^2}} \] while for solving the direction of vector we will us the formula which is \[\tan \theta = \dfrac{y}{x}\] in which \[x\] is the horizontal change and \[y\] is the vertical change

Complete step by step answer:
In the given equation \[2 - 4i\] firstly we will find the length of the vector.
\[\sqrt {{2^2} + {{( - 4)}^2}} \\
\Rightarrow 2\sqrt 5 \\ \]
Hence the length or magnitude of vector comes out to be \[\left| {\overrightarrow b } \right| = 2\sqrt 5 \]
Now in order to calculate the direction of \[2 - 4i\] we will angle \[\theta \] in CCW which means Counterclockwise direction with positive \[x\] axis we will get
\[\theta = - {\tan ^{ - 1}}\left| { - \dfrac{4}{2}} \right| \\
\Rightarrow\theta = - {\tan ^{ - 1}}(2) \\
\therefore \theta = - {63.435^ \circ } \\ \]
Hence, the vector has direction of \[ - {63.435^ \circ }\].

Note:Remember while solving the above equation that calculating the length is also known as magnitude which is shown by two vertical bars on either side of the vector. So while finding the length of vector \[x\] and \[y\] in the above equation we will write it as \[\left| {x,y} \right|\] and direction can be found out using inverse tangent function respectively.The length vector is often termed as magnitude and direction of vector is angle between horizontal axis and vector.