Orthonormal and orthogonal vectors

**mybrohshi5** · April 16th 2010, 02:10 PM

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

**Giraffro** · April 16th 2010, 06:36 PM

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 $\langle V_4, V_4 \rangle = 1$ .

**mybrohshi5** · April 17th 2010, 10:53 AM

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

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

I am having a brain fart when thinking about how to go about this

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

**Giraffro** · April 17th 2010, 07:04 PM

Originally Posted by mybrohshi5

I am a little confused

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

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

I am having a brain fart when thinking about how to go about this

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

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.

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