Component of $x$ that is orthogonal to $y$?

Question

suppose we have two vectors $x$, $y$, where $x$ has higher dimension than $y$.

What does it mean exactly to say that $w$ is “the component of $x$ that is orthogonal to $y$”?

Answer 1

Recall that given 2 vectors x and y we can consider the two components for x

• $x_{\parallel y}=\left(x\cdot \frac{y}{|y|}\right) \frac{y}{|y|}$ along $y$

• $x_{\perp y}=x-\left(x\cdot \frac{y}{|y|}\right) \frac{y}{|y|}$ orthogonal to $y$