Question

Express the vector r~ a b c d p r in terms of a~ , b~ , c~ , and d~ , the edges of a parallelogram.

Answer 1

Given the vector $\vec{r}(a,b,c,d)$.

Now taking the vertices of parallelogram asA $\vec{a},B\vec{b},C\vec{c},D\vec{d}$.

As we know to find the edges of parallelogram ABCD,we proceed as follows

$\vec{AB}$= Position vector of B - Position vector of A

                             $\vec{b}-\vec{a}$

$\vec{BC}$= Position vector of C - Position vector of B

                            =$\vec{c}-\vec{b}$

$\vec{CD}$= Position vector of D - Position vector of C

                           = $\vec{d}-\vec{c}$

$\vec{DA}$= Position vector of A - Position vector of D

                       =$\vec{a}-\vec{d}$

So, $\vec{b}-\vec{a},\vec{c}-\vec{b},\vec{d}-\vec{c},\vec{a}-\vec{d}$ are edges of parallelogram.