Question

a vector perpendicular to any other vector in the plane x+y+z=5 is​

Answer 1

Answer:

A plane is determined by a point on the plane and a vector orthogonal to the plane. Say P0 is a point on this plane, and n⃗ is the orthogonal vector. Also let's show the position of point P0 with the vector r0→.

Now take a generic point on the plane, call it P, show its position with the vector r⃗ . Then the following equation has to be satisfied:

n⃗ .(r⃗ −r0→)=0

This has to hold because the difference vector r⃗ −r0→ has to lie in our plane.

Now let's try to find out the scalar equation for our plane. First start by substituting our vectors:

Expanding our vector multiplication n⃗ .(r⃗ −r0→)=0 will give us:

a(x−x0)+b(y−y0)+c(z−z0)=0

If you define d=x0+y0+z0, then the previous equation becomes:

ax+by+cz=d.

From this equation you can identify our normal vector n⃗ directly from the coefficients ⟨a,b,c⟩ of the scalar equation of plane.