# The position vectors of the points A, B, C of a parallelogram ABCD are a, b, and c respectively. How do I express, in terms of a, b and, the position vector of D?

Aug 5, 2018

$a + c - b$.

#### Explanation:

Suppose that, the position vector (pv) of the point $D$ is $d$.

We dnote this by $D = D \left(d\right)$.

Now, we know from Geometry that, the diagonals $A C$ and $B D$

of a parallelogram $A B C D$ bisect each other.

Therefore, the mid-point of the diagonal $A C$ is the same as that of

the diagonal $B D$.

But, the pv. of $A C$ is $\frac{a + c}{2}$, &, that of $B D , \frac{b + d}{2}$.

$\therefore \frac{a + c}{2} = \frac{b + d}{2}$.

Clearly, $d = a + c - b$.

$\textcolor{v i o \le t}{\text{Enjoy Maths.!}}$