Find the unit vector, which is perpendicular to 2 vectors.

Question

How to solve the following problem:

You're given 2 vectors:

$$ \begin{equation} \alpha = (2; 10; 3) \end{equation} $$ $$ \begin{equation} \beta = (1; 4; 1) \end{equation} $$

Find the unit vector, which is perpendicular to these 2 vectors.

Answer 1

What you should do is apply the cross product to the two vectors,

$$\alpha \times \beta $$

The result will be perpendicular to the other two.

If you need a unit vector, you can always scale it down.

Answer 2

If you are cross multiplying two vectors then you will always find a vector which will have direction perpendicular to plane of the latter two (since the new vector is perpendicular to plane of those vectors so it will be perpendicular to each one of them).

In you case, you want to find

The unit vector perpendicular to vectors α=(2;10;3) and β=(1;4;1).

So for a vector which is perpendicular to α and β, cross multiply α and β. And make it unit vector by dividing it by the magnitude of resultant vector(αXβ).

Hope it help you

Answer 3

Solve the following underdetermined linear system

$$\begin{bmatrix} 2 & 10 & 3\\ 1 & 4 & 1\end{bmatrix} \mathrm x = \begin{bmatrix} 0\\ 0\end{bmatrix}$$

The solution set is a line passing through the origin. Intersect it with the unit Euclidean sphere.