# Thread: Can someone explain this to me?

1. ## Can someone explain this to me?

Find the values of x such that the given vectors are orthogonal. (Enter your answers from smallest to largest.)
xi + 3xj
xi - 5j

x = _______ (smaller value)
x = _______ (larger value)

The answer is 0 and 15, but how do I get that?

2. Let's consider the two vectors as two complex numerrs...

$\displaystyle z_{1} = x + 3 x i$

$\displaystyle z_{2} = x - 5i$

... where $\displaystyle i = \sqrt{-1}$ is the 'imaginary unit'. The condition of orthogonality is...

$\displaystyle Re\{\frac{z_{1}}{z_{2}}\} = 0$

... that leads at the algebraic equation...

$\displaystyle x^{2} - 15 x=0$

... whose solution are $\displaystyle x=0$ and $\displaystyle x=15$ ...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. They are orthogonal if, when you dot them, you get zero. So that gives you an equation to solve for x:

$\displaystyle (xi + 3xj)\cdot (xi - 5j)=x^2-15x=0$

Now solve that by factoring.

4. Thanks guys!