Find a point 90° left or right from a point (x,y,z) in a 3D space.

Question

How can I find a point which is 90° left or right from a point (x,y,z) in a 3D space?

for example if I have the point $(x,y,z)$ how to find $(x1,y1,z1)$ and $(x2,y2,z2)$.

Answer 1

Consider an arbitrary point $(x_1,y_1,z_1)$ and the vector $v=(x-x_1,y-y_1,z-z_1)$

In $\Bbb{R^3}$ there are infinitely many vectors $u=(x',y',z')$ satisfying $u\cdot v=0$

To find one of these, and thus a point with the desired property, we have $u\cdot v=0\Rightarrow x'(x-x_1)+y'(y-y_1)+z'(z-z_1)=0$.

So if for example you have $2x'-3y'+6z'=0$, then one such point would be $(2,1,\frac16)$ and another $(3,0,1)$.

PS Left or Right is meaningless in $\Bbb{R^3}$.

Answer 2

There is no right or left in a three dimensional world. With reference to the present direction of motion you have three degrees of freedom, pitch, roll or yaw. You need to roll right or left by $90^0.$

You can apply Euler angles: $ \theta ,\phi, \psi$ to a rigid body by means of a matrix multiplication populated mostly with $(1,-1,0) $. References and details are available on net.