Section Formula in 3D

The position vectors of two points A and B are$\overrightarrow{\mathrm{OA}}=2\hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}}$   and $\overrightarrow{\mathrm{OB}}=2\hat{\mathrm{i}}-\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ , respectively. The position vector of a point P which divides the line segment joining A and B in the ratio 2 : 1 is _____.

$P$ is a point on the line segment joining the points $(3,2,-1)\text{ and }(6,2,-2)$. If $x$- coordinate of $P$ is $5$ then find its $y$- coordinate.

Find the ratio in which the line segment joining the points $\left(4, 8, 10\right)$ and $\left(6, 10, -18\right)$ is divided by the $YZ$-plane.

Find the coordinates of the centroid of the triangle whose vertices are $\left(x_1, y_1, z_1\right), \left(x_2, y_2, z_2\right)$ and $\left(x_3, y_3, z_3\right)$.

Find the midpoint $M$ of the given points:

$A(-1,5,-4.2)$ and $\mathrm{B}(7,8.5,-11)$

If $D, E, F$ are midpoints of sides $BC, CA$ and $AB$ respectively of triangle $ABC$, then prove that $\overline{DG} + \overline{EG}+\overline{FG} = \overline{0}$.

If $ABC$ is a triangle whose orthocenter is $P$ and the circumcenter is $Q$, then prove that $\overline{PA}+\overline{PC}+\overline{PB}=\overline{PQ}$.

If $A, B, C, D$ are four non-collinear points in the plane such that $\overline{AD} + \overline{BD} + \overline{CD} = \overline{0}$, then prove that the point $D$ is the centroid of the triangle $ABC$.

If $G_1$ and $G_2$ are centroids of the triangles $ABC$ and $PQR$ respectively, then prove that $\overline{AP}+\overline{BQ}+\overline{CR}=3{\overline{G}}_1\overline{G_2}$.

Find the position vector of a point $R$  which divides the line joining two points $P$ and $Q$  whose position vectors are $P(\hat{i}+2\hat{j}-\hat{k})$ and $Q(-\hat{i}+\hat{j}+\hat{k})$ respectively, in the ratio $2 : 1$   externally

Find the position vector of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $P(\hat{i}+2\hat{j}-\hat{k})$ and $Q(-\hat{i}+\hat{j}+\hat{k})$ respectively, in the ratio $2 : 1$  internally

The coordinates of points of trisection of the line joining the points $A(x_1,y_1,z_1) \& B(x_2,y_2,z_2)$ are the points which divide $A \& B$ in the ratio $1:3 \& 3:1$ externally.

Find the coordinates of points of trisection of line joining the points $(2,3,4) \& (1,2,7)$

Using vector method, if $Q$ is the point of concurrence of the medians of the triangle $ABC$, then prove that $\overline{QA}+\overline{QB}+\overline{QC}=\overline{0}$.

Using vector method, prove that the centroid of the triangle formed by joining the mid points of the sides of a given triangle coincides with the centroid of the given triangle.

If $A(2,-2, 3), B(x, 4,-1), C(3, x,-5)$ are the vertices and$G(2, 1,-1)$ is the centroid of the triangle $ABC$, then, by vector method find the value of $x$.

 If $G(a, 2,-1)$is the centroid of the triangle with vertices $P(1,3,2)$ and $Q(5,b,-4)$and$R(5,1,c)$, then find the values of $a, b$ and $c$.
 

If the origin is the centroid of the triangle whose vertices are $A(2,p,-3), B(q,-2, 5)$and $C(-5,1, r)$  then find the values of $p, q$and $r$.

Find the position vector of the midpoint of the vector joining the points $P(2,3,4)$ and $Q(4,1,-2)$

Find the third vertex of triangle whose centroid is origin and two vertices are $(2, 4,6)$ and $(0,–2,–5)$.