Mathematics

Problem

Consider two non-zero, non-parallel vectors

\vec a

and

\vec b

in three–dimensional space. a. (i) Explain why

\vec a \times \vec b

is perpendicular to both

\vec a

and

\vec b

. b. (ii) Starting from the geometric definition of the cross product, derive an expression for the area of the parallelogram determined by

\vec a

and

\vec b

. c. (iii) Hence, show that the area of the triangle having

\vec a

and

\vec b

as two of its sides is

\dfrac12\,|\vec a \times \vec b|