Prove that if B.N = 0 and A is on the plane {p:(p – po).N = 0} then the entire line {p:p = A + tB} lies in the plane.

Answer 1

To prove that the entire line {p:p = A + tB} lies in the plane {p:(p - po).N = 0}, we need to show that for any value of t, the point A + tB satisfies the equation (p - po).N = 0.

Let A + tB be any point on the line. We can write this point as p = A + tB.

Now, let's substitute this value of p into the equation of the plane:

(p - po).N = 0

(A + tB - po).N = 0

Expanding the dot product, we have:

(A - po).N + t(B.N) = 0

Since B.N = 0 (given), the equation simplifies to:

(A - po).N = 0

But we also know that A is on the plane, so we can substitute A for p:

(A - po).N = 0

Since this equation holds for any value of t, we can conclude that the entire line {p:p = A + tB} lies in the plane {p:(p - po).N = 0}.