# Thread: Orthonormal and orthogonal vectors

1. ## Orthonormal and orthogonal vectors

Let

$V_1 = \begin{bmatrix}-0.5\\-0.5\\0.5\\-0.5 \end{bmatrix}$ $V_2 = \begin{bmatrix}-0.5\\0.5\\0.5\\0.5 \end{bmatrix}$ $V_3 = \begin{bmatrix}0.5\\0.5\\0.5\\-0.5 \end{bmatrix}$

Find a vector in such that the vectors , , , and are orthonormal.

$V_4 = \begin{bmatrix}.\\.\\.\\.\end{bmatrix}$

I know orthogonal vectors are those that if multiplied together equal 0.

I know that orthonormal vectors are vectors that are orthogonal and have a unit length of 1.

So $V_4$ will have to be able to be multiplied by $V_1 or V_2 or V_3$ and equal 0 in order to be orthogonal and then it must have a unit length of 1.

I just am not sure how to solve this type of problem

Thank you for any help

2. Let $V_4 = (a,b,c,d)^t$ and solve the simultaneous linear equations $\langle V_1, V_4 \rangle = \langle V_2, V_4 \rangle = \langle V_3, V_4 \rangle = 0$. You should then have $a,b,c,d$ in terms of one of those variables, $a$ say. Then solve $\langle V_4, V_4 \rangle = 1$.

3. Originally Posted by Giraffro
Let $V_4 = (a,b,c,d)^t$ and solve the simultaneous linear equations $\langle V_1, V_4 \rangle = \langle V_2, V_4 \rangle = \langle V_3, V_4 \rangle = 0$. You should then have $a,b,c,d$ in terms of one of those variables, $a$ say. Then solve $\langle V_4, V_4 \rangle = 1$.

I am a little confused

Can you get me started on how to solve the simultaneous linear equations

$\langle V_1, V_4 \rangle = \langle V_2, V_4 \rangle = \langle V_3, V_4 \rangle = 0$

This is what i have so far.

$V_1*V_4 ---> \begin{bmatrix}-0.5\\-0.5\\0.5\\-0.5\end{bmatrix} * \begin{bmatrix}a\\b\\c\\d\end{bmatrix} = 0$

$-\frac{1}{2}a + -\frac{1}{2}b + \frac{1}{2}c + -\frac{1}{2}d = 0$

$-1a + -1b + 1c = d$

I am not sure what to do from here now or if this is even right so far

4. Originally Posted by mybrohshi5
I am a little confused

Can you get me started on how to solve the simultaneous linear equations

$\langle V_1, V_4 \rangle = \langle V_2, V_4 \rangle = \langle V_3, V_4 \rangle = 0$

$V_1*V_4 ---> \begin{bmatrix}-0.5\\-0.5\\0.5\\-0.5\end{bmatrix} * \begin{bmatrix}a\\b\\c\\d\end{bmatrix} = 0$
$-\frac{1}{2}a + -\frac{1}{2}b + \frac{1}{2}c + -\frac{1}{2}d = 0$
$-1a + -1b + 1c = d$
You now have one variable in terms of the other 3, so carry on with $\langle V_2, V_4 \rangle = -\frac{a}{2} + \frac{b}{2} + \frac{c}{2} + \frac{d}{2}$ and substitute in $-a-b+c$ for $d$. Then find one variable in terms of the other 2 and so on.