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?

Question

Avery Allen · Accepted Answer

# a+c-b#.

Suppose that, the position vector (pv) of the point #D# is #d#. We dnote this by #D=D(d)#. Now, we know from Geometry that, the diagonals #AC# and #BD# of a parallelogram #ABCD# bisect each other. Therefore, the mid-point of the diagonal #AC# is the same as that of the diagonal #BD#. But, the pv. of #AC# is #(a+c)/2#, &, that of #BD, (b+d)/2#. #:. (a+c)/2=(b+d)/2#. Clearly, #d=a+c-b#. #color(violet)("Enjoy Maths.!")#

Autumn Armstrong · Answer

undefined The position vector of point D in terms of the position vectors of points A, B, and C in a parallelogram ABCD can be expressed using vector addition and subtraction. Since a parallelogram's opposite sides are parallel and equal in length, the vector representing the displacement from A to D will be the same as the displacement from B to C. Therefore, we can express the position vector of point D ($ \vec{D} $) in terms of the position vectors of points A ($ \vec{a} $), B ($ \vec{b} $), and C ($ \vec{c} $) as follows:

$$ \vec{D} = \vec{B} + \vec{C} - \vec{A} $$

$$ \vec{D} = \vec{b} + \vec{c} - \vec{a} $$