Distance and Section Formula in 3D

Find the volume and surface area of a regular tetrahedron with side $a=7 \mathrm{cm}$.

Find the volume and surface area of a regular tetrahedron with side $a=6 \mathrm{cm}$.

Find the volume and surface area of a regular tetrahedron with side $a=5 \mathrm{cm}$.

Find the volume and surface area of a regular tetrahedron with side $a=4 \mathrm{cm}$.

Find the volume and surface area of a regular tetrahedron with side $a=4 \mathrm{cm}$.

Find the volume and surface area of a regular tetrahedron with side $a=3 \mathrm{cm}$.

Find the volume and surface area of a regular tetrahedron with side $a=3 \mathrm{cm}$.

The points $\mathrm{A}(2,3,4),\mathrm{B}(-1,2,-3)$ and $\mathrm{C}(-4,1,-10)$ are collinear.

Verify whether the four points $\left(5, -1, 1\right), \left(-1, -3, 4\right),$ $\left(7, -4, 7\right)$ and $\left(1, -6, 10\right)$ are the vertices of rhombus or not.

Show that the points $\left(0, 7, 10\right), \left(-1, 6, 6\right)$ and $\left(-4, 9, 6\right)$ form a right angled triangle.

Show that points $\left(2, -1, -1\right), \left(4, -3, 0\right)$ and $\left(0, 1, -2\right)$ are collinear.

Find the values of $\lambda$ for which the points $(6, -1, 2),(8, -7, \lambda )$ and $\left(5, 2, 4\right)$ are collinear.

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

$ABCD$ is a parallelogram. $L$ is a point on $BC$ which divides $BC$ in the ratio$1:2$ . $AL$ intersects $BD$ at $P.M$ is a point on $DC$ which divides $DC$  in the ratio $1:2$ and $AM$ intersects $BD$ in $Q$.

Point $Q$ divides $DB$ in the ratio

$ABCD$ is a parallelogram. $L$ is a point on $BC$ which divides $BC$ in the ratio$1:2$ . $AL$ intersects $BD$ at $P.M$ is a point on $DC$ which divides $DC$  in the ratio $1:2$ and $AM$ intersects $BD$ in $Q$.

 Point $P$ divides $AL$ in the ratio

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 $(2,3,4) \& (1,2,7)$

The position vectors of the points $A$ and $B$ are $2\stackrel{\wedge }{i}-\stackrel{\wedge }{j} +5\stackrel{\wedge }{k}$and $-3\stackrel{\wedge }{i}+ 2\stackrel{\wedge }{j}$ respectively, find the position vector of the point which divides the line segment $AB$internally in the ratio $1:4$.

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