Need help, test coming soon!

Hello (Hi)

Since I don't know much about span, I'll take the easy one (Wink)

Quote:

Originally Posted by aeubz

Describe all vectors v = [x,y] that are orthogonal to u = [a,b].

Take the dot product of $\bold v$ and $\bold u$

$\bold u\cdot\bold v=ax+by$

The vectors are orthogonal when $\bold u\cdot\bold v=0$

So , we see that $ax+by=0\implies by=-ax$

Now, we need to manipulate the vector $\bold v$ :

$\left[\begin{array}{c}x\\y\end{array}\right]={\color{red}\frac{b}{b}}\left[\begin{array}{c}x\\y\end{array}\right]=\frac{1}{b}\left[\begin{array}{c}bx\\by\end{array}\right]$

Since, $by=-ax$ , $\bold v=\frac{1}{b}\left[\begin{array}{c}bx\\-ax\end{array}\right]=\frac{x}{b}\left[\begin{array}{c}b\\-a\end{array}\right]$

Since $\frac{x}{b}$ is another constant, let's denote it by $k$

So all the possible vectors will take on the form $\color{red}\boxed{\bold v=k\left[\begin{array}{c}b\\-a\end{array}\right];~k,a,b\in\mathbb{R}}$ .

Does this make sense? :D

--Chris