I need some help with this.

Question

How do I find a positively oriented orthonormal basis $ e_1, e_2, e_3$ such that $e_1, e_2$ are parallell to the plane $x + 2y + 4z = 7$ ? Can I use the normal of the plane and say that is $e_3$ and then that $ e_1 and \thinspace e_2$ is parallell to it?? How do I tackle this problem? The problem is in linear algebra.

Answer 1

Hint:

The vector normal to the plane is $\vec u=(1,2,4)^T$.

Chose two vectors on the plane, e.g. $\vec a=(1,1,1)^T$ and $\vec b=(1,0,3/2)^T$ (note that the components verify the equation of the plane) and take $\vec v=\vec a-\vec b$ that is parallel to the plane, so orthogonal to $\vec u$.

Take another vector $\vec w=\vec u \times \vec v$. ( that, by construction, is orthogonal to $\vec u$ and $\vec v$)

Normalize the three vectors and you have an orthonormal basis: $$ \vec e_1=\dfrac{\vec w}{|\vec w|} \qquad \vec e_2=\dfrac{\vec v}{|\vec v|} \qquad \vec e_3=\dfrac{\vec u}{|\vec u|} $$ with $\vec e_1$ and $\vec e_2$ parallel to the plane