A plane in space is determined by knowing a point on the plane and its "tilt" or orientation. This "tilt" is defined by specifying a vector that is perpendicular or normal to the plane. What is meant by the term "determined" here?
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3It means: The set of points which define the plane in $3-$space can be found if you have that data. – lulu Aug 09 '22 at 00:42
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3There is one and only one plane having that data. – GEdgar Aug 09 '22 at 00:44
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If you have a normal $n$ to a plane and a point $p$ which is part of the plane, let $q$ be any point in space, then $n\cdot(q-p)$ will be the scalar distance of the point $q$ from the plane. If the point $q$ is part of the plane then its scalar distance from the plane is zero, $n\cdot(q-p)=0$. Thus the normal $n$ and a point of the plane $p$ completely determine the plane, in other words we do not need any more information to decide whether $q$ is a point of the plane.
Exactly the same thing applies in two dimensions. If $n$ is a normal to a line and $p$ is a point on the line, then $n\cdot(p-q)=0$ is the equation of the line, so the line is completely determined by the normal and one point.
Geometrically, the normal tells you everything about the orientation of the plane or line, and then the point uniquely specifies which of all of the parallel planes or lines one means.
Suzu Hirose
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