Find the Magnitude,
Direction and Polar Form of a Vector

An interactive step by step calculator and solver to find the magnitude, direction and polar form of a vector is presented. As many examples as needed may be generated interactively including solutions with detailed explanations.

1) The magnitude $||\vec{v}||$ of a vector $ \vec{v} $ given by its components as $\vec{v} = \langle a, b \rangle $ is given by $||\vec{v}||=\sqrt{a^2+b^2}$
2) The direction of vector $\vec{v}$ may be defined as the angle $\theta$ made by the positive x-axis and the vector in counterclockwise direction such that $\tan(\theta) = \dfrac{b}{a}$.
3) The reference angle $\theta_r$ is the acute angle made by the x-axis and the vector, such that $\tan \theta_r = \left| \dfrac {b}{a} \right|$
Solve each step below then click on "Show me" to check your answer. There is a graph at the bottom of the page that helps you further understand graphically the solution to the question shown below.



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