Calculating an orthonormal vector from a given set of vectors in $ℝ^{3}$

Question

The following question below asks to find a vector $w$ and then a vector $u_{3}$ given a set of vectors and the information provided below. The vector calculations I can manage, but I seem to be getting tripped up on the orthonormal condition that the question asks for. Any advice or tips on approaching this problem would be highly appreciated.

Given the vectors; $$ u_{1}=\frac{1}{\sqrt{3}} \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix},\ u_{2}=\frac{1}{\sqrt{6}} \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix},\ v= \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$$

Calculate the vector

$$ w = v − \langle v,u_{1} \rangle u_{1} - \langle v,u_{2} \rangle u_{} $$

and hence find a vector $u_{3} ∈ R^{3}$, such that $ \{ u1, u2, u3 \} $ is an orthonormal set of vectors.

Answer 1

If you have computed $w$, then you're almost done: the set$$\left\{u_1,u_2,\frac w{\lVert w\rVert}\right\}$$will be an orthonormal set of vectors.

Answer 2

There is a typo in the text of the Problem/Exercise you posted which might confuse but probably its trivial for most

It should say:

Calculate the vector $$w=v-<u_{1},v>u_{1}+<u_{2},v>u_{2}$$.

The index 2 has been forgotten in the original text after the last Inner product in the expression for $w$

Other than that its exactly what poster Jose already said.

The vector w is orthogonal to both vectors $u_{1},u_{2}$(in fact orthogonal to the subspace spanned by $u_{1},u_{2}$) and you have nothing more to do than to normalize w in order to make your system orthonormal.

In case you have troubles or feel confusing ,the notion of orthogonality for vectors and how to check it

Consider the following facts/steps.

The vectors $u1,u2$ are normalized and orthogonal to one another.

Note that the vector spaces you consider when you ask for orthogonality are usualy spaces where the norm is defined via an Inner-Product(for real numbers as scalars this means symmetric,postive definite bilinearform) in the following sense $$||v||:=<v,v>^{\frac{1}{2}}$$

To check if 2 vectors x,y are orthogonal you have to check for $<x,y>=0$.

First

If a vector v is projected on a normalized vector u by

$P(v)=<u,v>u$ ,the vector $v-P(v)$ will be orthogonal on u.

This can be easily verified by straightforward calculation checking orthogonality condition for $x=v-P(v)$ and $y=u$

Second

If a vector w is orthogonal to the unit vectors u1,u2 then w is orthogonal to all vectors spanned by these.

Again this just straigth forward calculation expressing an arbitrary vector b of $span\{u_1,u_2\}$ as a Linearcombination of $u_{1},u_{2}$ and using inner product properties.

Last thing important to know is that vectors which are pairwise orthogonal to one another

are linear independent.

When you do these calculations just keep in mind the symmetry and linearity properties of the Inner product(<,>) and how the Inner product and Norm of your space are related and everything works out smoothly.