Projecting a vector on orthogonal planes

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I am looking from an engineer point of view. I have a sensor for which I need vector projecting on two different planes. I have the unit vector in the body frame that is to be projected and I obtained that using the rotation matrix. Also I have the unit vectors of the frame which is formed by the three planes. If {n1, n2, t} represents those three unit vectors and s gives me the unit vector measured in body frame, how can I obtain projection of s on n1,t and n2,t plane.

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If the vectors n1, n2, and t are mutually orthogonal, the projection of s onto the n1,t plane is "s - (sn2)n2", that is, find the projection of s onto n2, the normal to the n1,t plane, and subtract it from s. Similarly, the projection onto the n2,t plane would be "s - (sn1)n1". This is not a change of coordinate system, if that is what you wanted. If that is what you wanted, you can obtain the coordinates in the n1, n2 and t directions by dotting the unit vectors with s, i.e. "s = s_i * e_i' = s * e_i" where s_i is the component of s in the e_i direction and e_i belongs to {n1,n2,t}.