Section Formula

Show that the points $O(0,0,0), A(2,-3,3), B(-2,3,-3)$ are collinear. Find the ratio in which each point divides the segment joining the other two.

$A(3,2,0), B(5,3,2), C(-9,6,-3)$ are vertices of a triangle. $\overline{AD}$, the bisector of $\angle BAC$ meets $\overline{BC}$ at $D$. Find the coordinates of $D$.

Find the point of intersection of the lines $\overrightarrow{AB}$ and $\overrightarrow{CD}$ where $A=(7,-6,1), B=(17,-18,-3)$, $C=(1,4,-5)$ and $D=(3,-4,11)$.

If $A(4,8,12), B(2,4,6), C(3,5,4)$ and $D(5,8,5)$ are four points, show that the lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ intersect.

Show that the points $A(3,2,-4), B(5,4,-6)$ and $C(9,8,-10)$ are collinear and find the ratio in which $B$ divides $\overline{AC}$.

If $(3,2,-1), (4,1,1)$ and $(6,2,5)$ are three vertices and $(4,2,2)$ is the centroid of a tetrahedron, find the fourth vertex.

Find the coordinates of the vertex $C$ of $∆ABC$ if its centroid is the origin and the vertices $A$, $B$ are $(1,1,1)$ and $(-2,4,1)$ respectively.

Find the ratio in which the $XZ-$plane divides the line joining $A(-2,3,4)$ and $B(1,2,3)$.