Vectors The vectors U V and W shown below with bold lines fr

Vectors. The vectors U, V, and W (shown below with bold lines) from three sides of a quadrilateral. The vectors A, B, C, and D (shown with dotted lines) join the midpoints of the quadrilateral. Express the fourth side of the quadrilateral in terms of U, V, and W. Express the vectors A, B, C, and D in terms of U, V, and W A = B = C = D = What does this say about the quadrilateral formed by joining the midpoints?

Solution

For the quadrilateral, u+v+w+x =0 where x is the fourth side

since the starting point and end points are the same

Hence x = fourth side = -u-v-w

A =(w+u)/2

B =(x-w)/2 = -u-v-w+w/2 = -(u+v)/2

C = (-v-x)/2 = (u+w)/2

and D = (u+v)/2

Hence opposite sides are parallel.

ABCD is a parallelogram