Question

Consider the three linearly independent vectors \[{\vec{\boldsymbol u}}=\frac{ 1 }{ \sqrt{3} }\left(\begin{matrix}1 \\ 1\\ 1\end{matrix}\right), {\vec{\boldsymbol v}}=\frac{ 1 }{ \sqrt{6} }\left(\begin{matrix}2 \\ -1\\ -1\end{matrix}\right), {\vec{\boldsymbol w}}=\frac{ 1 }{ \sqrt{2} }\left(\begin{matrix}0 \\ 1\\ -1\end{matrix}\right)\]

Show that they form an orthonormal basis.

Answer 1

I know that for the basis to be orthonormal I need u·u=v·v=w·w=1 and u·v=u·w=
w·v = 0, but I'm not sure how to show this. I tried multiplying each vector element by the scalar, but then how would I multiply e.g. u·u or u·v? Maybe I am misunderstanding something about dot products and matrix multiplication. Any help is very much appreciated.

Answer 2

I don't know much about orthonormal basis, It seems like this is a problem about dot product

Your representation is similar to this,
$$\vec {u}=\frac{1}{\sqrt{3}}\langle 1,1,1\rangle$$$$\vec {v}=\frac{1}{\sqrt{6}}\langle 2,-1,-1\rangle$$$$\vec {w}=\frac{1}{\sqrt{2}}\langle 0,1,-1\rangle$$
So,
$$\vec{u}\cdot\vec v=\frac{1}{\sqrt{3}}\langle 1,1,1\rangle\cdot \frac{1}{\sqrt{6}}\langle 2,-1,-1\rangle=\frac{1}{3\sqrt{2}}(2\times 1,1\times -1,1\times -1)=0$$

I hope this is helpful

Answer 3

Sorry I made a little mistake in the last line, it should be
$$\vec{u}\cdot\vec v=\frac{1}{\sqrt{3}}\langle 1,1,1\rangle\cdot \frac{1}{\sqrt{6}}\langle 2,-1,-1\rangle=\frac{1}{3\sqrt{2}}[(2\times 1)+(1\times -1)+(1\times -1)]=0$$

Answer 4

a few conditions have to hold; the vectors must be independant, orthogonal to each other, and have a unit length.

Answer 5

the scalar on the outside can be used to determine the unitness of them; you can determine the orthogonal and indepence without the scalar to make the numbers easier to manage

Answer 6

Thanks @BAdhi for the example, it was the addition in the last line you added that I forgot about and that kept throwing me off. Thanks also for the helpful tips @amistre64, hopefully I will get my head around all this soon.

Answer 7

it does get better, for me it was just trying to remember the terms applied to concepts that i already knew :)

good luck