# What does it mean for two vectors to be orthogonal?

Nov 7, 2015

Their dot product is equal to $0$.

#### Explanation:

It just means they are perpendicular. To find this, take the dot product by taking the first times first plus last times last. If this equals zero, they are orthogonal.

for example: $< 1 , 2 > \cdot < 3 , 4 > = \left(1 \cdot 3\right) + \left(2 \cdot 4\right) = 11$

This is also known as the inner product.

For 3D-vectors, do basically the same thing, including the middle term.

for example: $< 4 , 5 , 6 > \cdot < 0 , 1 , 2 > = \left(4 \cdot 0\right) + \left(5 \cdot 1\right) + \left(6 \cdot 2\right) = 17$

Think of two vectors, one pointing straight up, and one pointing straight to the right. Those vectors could be defined like so:
$< 0 , a >$ and $<$$b , 0$$>$
Since they form a right angle, they are orthogonal. Taking the dot product we find...
$< 0 , a >$$\cdot$$<$$b , 0$$> = \left(0 \cdot b\right) + \left(a \cdot 0\right) = 0$

Nov 7, 2015

Essentially, they are at right angles to one another and their dot product is zero.

#### Explanation:

If they are also of length $1$, then they are called orthonormal.

A set of $n$ orthonormal vectors in $n$ dimensional space is called an orthonormal basis.

If you form an $n \times n$ matrix $A$ whose rows are those vectors, then it is invertible, with inverse equal to its transpose. That is: ${A}^{- 1} = {A}^{T}$. You get the the result if you form a matrix whose columns are an orthonormal basis.

Such a matrix represents an orthogonal transformation - preserving angles and distances - essentially a combination of rotation and possible reflection.