= Orthogonal vectors
{title2=$\langle u,v\rangle=0$}
= Orthogonal
{synonym}
= Orthogonality
{synonym}
Two <vectors> are orthogonal when their <inner product> is zero.
For real <Euclidean vectors>, the <inner product> is the <dot product>, so $(3,4)$ and $(4,-3)$ are <orthogonal>: $3(4)+4(-3)=0$. For nonzero <Euclidean vectors>, this means their <angle> is $90^\circ$. The <zero vector> is <orthogonal> to every <vector>, although it has no direction or defined <angle>.https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/resources/lecture-14-orthogonal-vectors-and-subspaces/[Strang, MIT 18.06, Lecture 14: orthogonal vectors and subspaces]{ref}
In an abstract <inner product space>, <orthogonality> always refers to its chosen <inner product>. For example, the <weighted inner product> $\langle u,v\rangle_w=4u_1v_1+u_2v_2$ makes $(1,1)$ and $(1,-4)$ <orthogonal>, since $4-4=0$. Their ordinary <dot product> is $-3$, so their arrows are not perpendicular in the usual <Cartesian coordinate system>. The algebraic definition also applies to <functions>, where arrows need not be available.
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