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Vector Projection

linear algebra vector math
Table of Contents

For any vectors \(\small \vec{a}\) and \(\small \vec{b}\):

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The projected (red) and perpendicular (green) vector can be calculated as follows:

$$ proj_{\vec{b}}(\vec{a})=\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\vec{b}=(\vec{a}\cdot\hat{b})\hat{b} $$

$$ perp_{\vec{b}}(\vec{a})=\vec{a}-proj_{\vec{b}}(\vec{a}) $$

Where

  • \(\small \vec{a}\) is the vector to be projected
  • \(\small \vec{b}\) is the vector to be projected onto
  • \(\cdot\) is the dot product
  • \(\small |\vec{b}|^2\) is the magnitude of \(\small \vec{b}\), squared
  • \(\small \hat{b}\) is the unit vector in the direction of \(\small \vec{b}\)

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