In a tetrahedron

MATHEMATICS · Subjective · CBSE · Mathematics · Vector Algebra · Vector Algebra

Problem

In a tetrahedron

ABCD

, the position vectors of the vertices with respect to the origin are

\overrightarrow{OA}=\mathbf{i}+2\mathbf{k},\; \overrightarrow{OB}=4\mathbf{i}-\mathbf{j}+3\mathbf{k},\; \overrightarrow{OC}=2\mathbf{i}+2\mathbf{j}+5\mathbf{k}

and

\overrightarrow{OD}=3\mathbf{i}+\mathbf{j}+7\mathbf{k}

. (i) Obtain a unit vector perpendicular to the two edges

\overrightarrow{AB}

and

\overrightarrow{AC}

such that its

z

–component is positive. (ii) Using the vector obtained in part (i), calculate the perpendicular distance of vertex

D

from the plane

ABC

. (iii) Hence find the volume of tetrahedron

ABCD