Art Unwin wrote:
"I have just come to realize that if one drew a polygon of element
phases in an array and all elements were 180 degrees to its companion
element and excluding its driven element, the max gain and max front to
back will occur at the dame frequency!"

I missed a step or two between the polygon`s resultant and coincidence
of maximum gain with maximum front to back ratio.

A vector has both magnitude and direction. A scalar quantity has only
magnitude.

Vectors are represented by arrows whose lengths correspond to their
magnitudes. Directions of the arrows correspond to the directions of the
vectors.

The combined effect of two or more vectors is called a resultant. A
resultant can be found by a geometrical rule called vector addition.
Vectors are placed head to tail while maintaining their magnitudes and
directions. The resultant is a vector drawn from the tail of the first
vector to the head of the final vector. Any number of vectors can be
added by the head to tail method. These can create a polygon of vectors.

If only two vectors are to be added at a time, they can produce a
resultant by the parallelogram method. Two sides of the parallelogram
are formed by the two vectors connected at their tails. Parallel same
length sides are added to form the parallelogram. A diagonal emanating
from the junction of the two vectors forms the resultant

When one vector (call it c) is the resultant of two vectors (call them
and b) we can say that a and b are components of c. Any given vector can
be resolved into an infinite number of pairs. Usually we find it
convenient to resolve a vector into a pair which are at right angles
with each other. Then we can use the Pythagorean theorem (c squared = a
squared + b squared) to find the magnitude of the resultant (c).

In a right triangle in which two sides are perpendicular, all the
trignometric functions are useful in determining the lengths and
directions of its sides.

Best regards, Richard Harrison, KB5WZI