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Projection

0.1. Projection Projection is used to find the component of one vector in the direction of the vector that you are projecting onto.
The formula for projection is as follows:
\begin{equation}
\vectorproj[x]{y} = \frac{\vec{x}\cdot\vec{y}}{{||\vec{x}||}^2}\vec{x}
\end{equation}
The above equation takes in vectors and outputs a vector. Some questions require purely just the distance, and there is a shortcut formula for getting purely the magnitude.
\begin{equation}
||\vectorproj[x]{y}|| = |\frac{\vec{y}\cdot\vec{x}}{||\vec{x}||}|
\end{equation}
There is another component that can be found is the perpendicular part. Perpendicular part is perpendicular to the projection. When projection and perpendicular is added, the original vector that was projected is formed.
\begin{equation}
\begin{split}
\vectorproj[x]{y} + \vectorperp[x]{y} = \vec{y}
\vectorperp[x]{y} = \vec{y} – \vectorproj[x]{y}
\end{split}
\end{equation}

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