A plane in 3D can be defined using various combinations of
- points lying in the plane,
- direction vectors in (or parallel to) the plane, and
- normal vectors which are perpendicular to the plane.
For each of the following combinations of data,
- does it define a unique plane?
- does it define a family of possible planes? If so, can you describe the family?
The points \((-1,4,6)\), \((0,7,4)\) and \((3,-1,10)\)
The points \((-1,4,6)\), \((0,7,4)\) and \((-3,-2,10)\)
The direction vectors \(\mathbf{i}+3\mathbf{j}+\mathbf{k}\) and \(-2\mathbf{i}-6\mathbf{j}-2\mathbf{k}\)
The direction vectors \(\mathbf{i}+3\mathbf{j}\),
\(\mathbf{i}+\mathbf{j}+2\mathbf{k}\) and
\(2\mathbf{i}-\mathbf{j}\)
The direction vectors \(\mathbf{i}+2\mathbf{j}+\mathbf{k}\), \(-2\mathbf{i}-4\mathbf{j}+3\mathbf{k}\) and \(3\mathbf{i}+6\mathbf{j}+5\mathbf{k}\)
The point \((1,5,3)\) and direction vectors \(\mathbf{i}-3\mathbf{j}+2\mathbf{k}\) and \(5\mathbf{i}+2\mathbf{j}+2\mathbf{k}\)
The points \((1,5,8)\) and \((-2,17,3)\) and direction vector \(-3\mathbf{i}+2\mathbf{j}-\mathbf{k}\)
The points \((1,-6,2)\) and \((3,12,-4)\) and direction vector \(\mathbf{i}+9\mathbf{j}-3\mathbf{k}\)
The point \((7,-3,2)\) and normal vector \(-2\mathbf{i}+8\mathbf{j}-12\mathbf{k}\)
The normal vector \(-2\mathbf{i}+\mathbf{j}+4\mathbf{k}\) and points \((-3,3,4)\) and \((1,-1,7)\)
The normal vector \(-2\mathbf{i}+\mathbf{j}+4\mathbf{k}\) and points \((6,-3,5)\) and \((1,-1,7)\)
The direction vector \(4\mathbf{i}+8\mathbf{j}-10\mathbf{k}\) and normal vector \(3\mathbf{i}+\mathbf{j}+2\mathbf{k}\)
