question_answer

If the plane \[\frac{x}{2}+\frac{y}{3}+\frac{z}{4}=-\frac{1}{2}\] intersects \[x,y\] and z-axis at A, B and C respectively then volume of the tetrahedron OABC where '0' is the origin is

A)  6                                

B)  4

C)  3                                

D)  \[\frac{1}{2}\]

Correct Answer: D

Solution :

\[\frac{x}{2}+\frac{y}{3}+\frac{z}{4}=\frac{1}{2}\] \[V=\frac{1}{6}[\overrightarrow{OA}\,\,\,\overrightarrow{OB}\,\,\,\overrightarrow{OC}]\] \[=\frac{1}{6}\,\left| \,\left[ \hat{i}\frac{3}{2}\hat{j}\,\,2\hat{k} \right]\, \right|\] \[=\frac{1}{6}\cdot \frac{3}{2}\cdot 2=\frac{1}{2}\]