Find the Angle Between two Vectors

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An interactive step by step calculator and solver to find the angle between two vectors is presented. As many examples as needed may be generated along with their solutions and detailed explanations.

1) The angle $\theta$ between two vectors $\vec{u}$ and $\vec{v}$ is given by: $\theta= \arccos(\dfrac{\vec{u}\cdot \vec{v}}{||\vec{u}|| \cdot ||\vec{v}||})$
2) The magnitude $||\vec{u}||$ of a vector $ \vec{u} $ given by its components as $\vec{u} = \langle a, b \rangle $ is given by $||\vec{u}||=\sqrt{a^2+b^2}$
3) If $\vec{u} = \langle a, b \rangle $ and $\vec{v} = \langle c, d \rangle $ , the dot product $\vec{u}\cdot \vec{v} = a \cdot c + b \cdot d$
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.

Step by step solution

Below are shown the two vectors, $\vec u$ (green) $\vec v$ (blue) and angle $\theta, \text{such that } 0\le \theta \le 180^{\circ}$, (red) between them.

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