In space with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ we are given the points: $$\mathrm{A}(1;2;3),\ \mathrm{B}(3;0;1),\ \mathrm{C}(-1;0;1),\ \mathrm{D}(2;1;-1),\ \mathrm{E}(-1;-2;3)\ \text{and}\ \mathrm{F}(-2;-3;4).$$ For each statement, say whether it is true or false by justifying your answer. An unjustified answer will not be taken into account. Statement 1: The three points $\mathrm{A}$, $\mathrm{B}$, and C are collinear. Statement 2: The vector $\vec{n}(0;1;-1)$ is a normal vector to the plane (ABC). Statement 3: The line $(\mathrm{EF})$ and the plane $(\mathrm{ABC})$ are secant and their point of intersection is the midpoint of segment [BC]. Statement 4: The lines (AB) and (CD) are secant.
In space with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ we are given the points:
$$\mathrm{A}(1;2;3),\ \mathrm{B}(3;0;1),\ \mathrm{C}(-1;0;1),\ \mathrm{D}(2;1;-1),\ \mathrm{E}(-1;-2;3)\ \text{and}\ \mathrm{F}(-2;-3;4).$$
For each statement, say whether it is true or false by justifying your answer. An unjustified answer will not be taken into account.
Statement 1: The three points $\mathrm{A}$, $\mathrm{B}$, and C are collinear.\\
Statement 2: The vector $\vec{n}(0;1;-1)$ is a normal vector to the plane (ABC).\\
Statement 3: The line $(\mathrm{EF})$ and the plane $(\mathrm{ABC})$ are secant and their point of intersection is the midpoint of segment [BC].\\
Statement 4: The lines (AB) and (CD) are secant.