Distance of Point from Hyperplane: Self Test

Problem 1

For each of the statements below about a point \(P\) and hyperplane \(H\) in \(\RNrSpc{n}\) indicate if it is true or false

  1. \(P\) always lies on \(H\).

  2. \(P\) always has distance \(0\) from \(H\)

  3. The distance from \(P\) to \(H\) is \(0\) exactly when \(P\) lies on \(H\).

  4. The distance from \(P\) to \(H\) is only defined if any two points of \(H\) have the same distance from \(P\).

  5. The distance from \(P\) to \(H\) is the distance of \(P\) from all points of \(H\)

  6. The distance from \(P\) to \(H\) is the distance from \(P\) to the point of \(H\) closest to \(P\).

  7. If the distance from \(P\) to \(H\) is \(d\), then

    \(\Dstnc{P}{Q}\)\(\geq\)\(d\)

    for every \(Q\) on \(H\).

  8. If the distance from \(P\) to \(H\) is negative, then \(P\) lies on the other side of \(H\).

  9. If the distance from \(P\) to \(H\) is positive, then \(P\) does not lie on \(H\).