I'm having issues with this problem.

Question

With the formula for projection, we can tell that the projection of v onto the plane II with proj_IIv = v - proj_bv where n is a vector normal to II. If II is the plane with the equation ax + by + cz = d and v = <v1, v2, v3>, find a formula involving just the coefficients of the equation and the three components of v for the projection of the vector v onto the plane II. The formula should also tell how to compute each component of the projection.

Answer 1

note: * is multiplication not dot product

correction

The scalar projection of v onto n is [a*v1+b*v2+c*v3]/√(a²+b²+c²)

forgot to divide by magnitude of n.

the vector to subtract from v is the scalar times unit n

that is, (a*v1+b*v2+c*v3)*(a,b,c)/(a²+b²+c²)

corrected two mistakes, no square root on bottom, and used (a,b,c) as n

is subtracted from (v1,v2,v3) to find projection

of v onto plane parallel with ax+by+cz=d

so first element of vector is v1-a*(a*v1+b*v2+c*v3)/(a²+b²+c²)

second element v2-b*(a*v1+b*v2+c*v3)/(a²+b²+c²)

third element v3-c*(a*v1+b*v2+c*v3)/(a²+b²+c²)