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# Two orthogonal vectors $u$ and $v$ are given. Compute the quantities $||u||^2$, $||v||^2$, and $||u+v||^2$. Use your results to illustrate the Pythagorean theorem.$\begin{equation*} u=\begin{bmatrix}1\\3\\2\end{bmatrix}\text{ and } v=\begin{bmatrix}-1\\1\\-1\end{bmatrix} \end{equation*}$

Solution

Verified

$u = \begin{bmatrix} 1\\ 3\\ 2 \end{bmatrix}$

;

$v = \begin{bmatrix} -1\\ 1\\ -1 \end{bmatrix}$

$||u||^2 = 1^2+3^2+2^2= 14$

$||v||^2 = (-1)^2+1^2+(-1)^2= 3$

$||u+v||^2 =(1+(-1))^2 + (3+1)^2+(2+(-1))^2= 17 = ||u||^2+||v||^2$

So using Pythagorean theorem we conclude that the vectors $u$ and $v$ are orthogonal.

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