The Vector Projection

Suppose we have two vectors $\overrightarrow{X}$ and $\overrightarrow{Y}$ which point in different directions. In some scenarios, we might want to try and rewrite $\overrightarrow{X}$ as a combination of two component vectors $\overrightarrow{X}^\parallel$ and $\overrightarrow{X}^\perp$, with one of these parallel to $\overrightarrow{Y}$ and the other perpendicular.

We call the parallel component $\overrightarrow{X}^\parallel$ the vector projection or the vector resolution of $\overrightarrow{X}$ onto $\overrightarrow{Y}$. We can use the dot product to compute the vector projection through the formula

$\overrightarrow{X}^\parallel = \operatorname{Proj}_{\overrightarrow{Y}}\overrightarrow{X} = \left(\frac{\overrightarrow{X} \cdot \overrightarrow{Y}}{\overrightarrow{Y} \cdot \overrightarrow{Y}}\right) \overrightarrow{Y}$

We can then use this formula to find the perpendicular component using

$\overrightarrow{X}^\perp = \overrightarrow{X} - \overrightarrow{X}^\parallel$

We can also define the scalar projection or scalar resolution of $\overrightarrow{X}$ onto $\overrightarrow{Y}$, which is the length of the vector $\overrightarrow{X}^\parallel$. The formula for the scalar projection is also related to the dot product

$|\overrightarrow{X}^\parallel| = \frac{\overrightarrow{X} \cdot \overrightarrow{Y}}{|\overrightarrow{Y}|} = \frac{\overrightarrow{X} \cdot \overrightarrow{Y}}{\sqrt{\overrightarrow{Y} \cdot \overrightarrow{Y}}}$

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