$\overline{x} \times \overline{a} = \overline{b}$ has a solution when $ \langle\overline{a},\overline{b} \rangle =0$

Question

I'm trying to solve this exercise:

Let $\overline{a} \neq \overline{0}$, $\overline{b}$ be two vectors of the Euclidean vector space $V_{3}$. Prove the equation $\overline{x} \times \overline{a} = \overline{b}$ has a solution when $ \langle\overline{a},\overline{b} \rangle =0$ and find if it's the only one.

It does not seem that hard, my approach is the following, but I seem to be missing something:

Let $B = \left \{ \overline{\epsilon_{1}}, \overline{\epsilon_{2}}, \overline{\epsilon_{3}} \right \}$ be a base of $V_{3} $

Then the vectors $\overline{a}$. $\overline{b}$ can be represented as:

$\overline{a} = a_{1}\overline{\epsilon_{1}} + a_{2}\overline{\epsilon_{2}} + a_{3}\overline{\epsilon_{3}} = \sum_{i=1}^{3}a_{i}\overline{\epsilon_{i}}$

$\overline{b} = \sum_{j=1}^{3}b_{j}\overline{\epsilon_{j}}$

I also analyze the vector $\overline{x}$ as follows:

$\overline{x} = x_{1}\overline\epsilon_{1} + x_{2}\overline\epsilon_{2} +x_{3}\overline\epsilon_{3}$

$\overline{x} \times \overline{a} = \overline{b}$

$(x_{2}a_{3} - x_{3}a_{2})\overline\epsilon_{1} + (x_{3}a_{1} -x_{1}a_{3})\overline\epsilon_{2} + (x_{1}a_{2} - x_{2}a_{1})\overline\epsilon_{3} = \sum_{j=1}^{3}b_{j}\overline\epsilon_{j}$

And I get:

$x_{2}a_{3} - x_{3}a_{2} = b_{1}$

$x_{3}a_{1} - x_{3}a_{3} = b_{2}$

$x_{1}a_{2} - x_{3}a_{1} = b_{3} $

I am aware that $\langle\overline{a},\overline{b} \rangle =0$ means that the two vectors at vertical with each other, but how to I proceed from here?

Answer 1

$$ 0 \cdot x_1+ a_3\cdot x_2 -a_2\cdot x_3=b_1\\ -a_3\cdot x_1+0\cdot x_2+a_1\cdot x_3=b_2\\ a_2 \cdot x_1 -a_1 \cdot x_2 + 0 \cdot x_3=b_3\\ $$ This is a system of linear equation. We are given that the vector a is nonzero. Without loss of generosity, let us assume that a_3 is nonzero.

Then by using Gaussian elimination, we get:

$$ \begin{array}{ccc|c} 1 & 0 & -a_1/a_3 & -b_2/a_3 \\ 0 & 1 & -a_2/a_3 & b_1/a_3 \\ 0 & 0 & 0 & (a_1b_1+a_2b_2+a_3b_3)/a_3 \end{array} $$

From linear algebra, we know that this system only has solution iff $$ a_1b_1+a_2b_2+a_3b_3=0 $$ which is equivalent to the dot product is zero.

Answer 2

We know that $\vec x \times \vec a = \vec b$ means that $\vec a \perp \vec b$

If the two vectors are not perpendicular then there are no solutions. The following system has no solutions:

$$\left\{ \begin{gathered} {a_3}{x_2} - {a_2}{x_3} = {b_1} \\ {a_1}{x_3} - {a_3}{x_1} = {b_2} \\ {a_2}{x_1} - {a_1}{x_3} = {b_3} \\ \end{gathered} \right.$$

We know that $\vec a\perp\vec b$ which means $a_1b_1+a_2b_2+a_3b_3=0$

we can solve for $b_3$ getting $b_3=-\dfrac{a_1b_1+a_2b_2}{a_3}$ and substitute into the previous system

$$\left\{ \begin{gathered} {a_3}{x_2} - {a_2}{x_3} = {b_1} \\ {a_1}{x_3} - {a_3}{x_1} = {b_2} \\ {a_2}{x_1} - {a_1}{x_3} = -\dfrac{a_1b_1+a_2b_2}{a_3} \\ \end{gathered} \right.$$

Solving the system we get: $$\left\{ \begin{gathered} {x_1} = \frac{{{b_2}{b_3} - {a_1}{b_3}{x_3}}}{{{a_1}{b_1} + {a_2}{b_2}}} \\ {x_2} = - \frac{{{b_1}{b_3} - {a_2}{b_3}{x_3}}}{{{a_1}{b_1} + {a_2}{b_2}}} \\ {x_3} = {x_3} \\ \end{gathered} \right.$$

Which shows that the solution is not unique