Inner product of perpendicular and orthogonal vectors

Question

Perpendicular means two vectors have a 90-degree angle.
The inner product of perpendicular vectors is 0.
$ ab = |a||b|\cos\theta$, where $\theta = 90$, and $\cos\theta = 0$

But the inner product of two orthogonal matrices (sets of perpendicular vectors) is $I$ rather than 0, like $QQ^T = I$. Why?

Really confused. Thanks~

Answer 1

I think what is confusing you is that the adjectives "orthogonal" and "perpendicular" are synonyms in ordinary English. In mathematics their precise meanings depend on the context.

For nonzero vectors they mean the same thing.

Ab orthogonal matrix is a matrix whose columns (which are vectors) are orthogonal to each other (and each individual column has length $1$).

PS. The product of two matrices is not their "inner product".

Answer 2

If $Q$ is an orthogonal matrix and if the lines of $G$ are $v_1,v_2,\ldots,v_n$m then$$QQ^T=\begin{bmatrix}\langle v_1,v_1\rangle&\langle v_1,v_2\rangle&\ldots&\langle v_1,v_n\rangle\\\vdots&\vdots&\ddots&\vdots\\\langle v_n,v_1\rangle&\langle v_n,v_2\rangle&\ldots&\langle v_n,v_n\rangle\end{bmatrix},$$and this is the identity matrix since $\langle v_i,v_j\rangle=0$ if $i\neq j$ and it is equal to $1$ if $i=j$.

Answer 3

$QQ^T=[q_{ij}]=[r_i\cdot r_j]$, where $r_i,r_j$ represent the $i^{th}$ and $j^{th}$ row vectors of $Q$. Thus,$$q_{ij}=\begin{cases}r_i\cdot r_i=|r_i|^2=1,&i=j\\0,&i\ne j\end{cases}$$

Answer 4

It's $I$ rather than zero because when you take inner products in the matrix product, like $v_i \cdot v_j$, sometimes $i = j$, in which case the two vectors are not orthogonal --- they're identical, and the inner product is the squared length of the vector.