Vector Orthogonality

In geometry, we call two vectors $\mathbf x$ and $\mathbf y$ orthogonal of the angle between then is 90 - i.e. they are perpendicular.


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Inner Product Test

  • If $\mathbf x$ and $\mathbf y$ are perpendicular, then we can use the Pythagoras theorem
  • $\| \mathbf x \|^2 + \| \mathbf y \|^2 = \| \mathbf x + \mathbf y* \|^2$
  • is there an easier way to tell if 2 vectors are orthogonal?
  • yes! if their Inner Product is zero, then they are: $\mathbf x^T \mathbf y = \sum x_i y_i = 0 \Rightarrow \mathbf x \, \bot \, \mathbf y$


Why?

  • $\| \mathbf x \|^2 = \left(\sqrt{\sum x_i^2 } \right)^2 = \mathbf x^T \mathbf x$
  • let's expand the Pythagoras theorem:
  • $\| \mathbf x \|^2 + \| \mathbf y \|^2 = \| \mathbf x + \mathbf y* \|^2$
  • $\mathbf x^T \mathbf x + \mathbf y^T \mathbf y = (\mathbf x + \mathbf y)^T (\mathbf x + \mathbf y) = \mathbf x^T \mathbf x + \mathbf x^T \mathbf y + \mathbf y^T \mathbf x + \mathbf y^T \mathbf y$
  • or $\mathbf x^T \mathbf y + \mathbf y^T \mathbf x = 0$
  • note that $\mathbf x^T \mathbf y = \mathbf y^T \mathbf x$, so we have
  • $2 \mathbf x^T \mathbf y = 0$ or $\mathbf x^T \mathbf y = 0$


Zero Vectors

Zero vectors $\mathbf 0$ are orthogonal to any vector in its space

  • $\mathbf 0 \; \bot \; \mathbf x$ $\forall \mathbf x$
  • because $\mathbf 0^T \mathbf x = 0$



Sources

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