[3.3] (Division of a segment into congruent parts) Let n be any integer greater than 1. Let α and β be numbers. If β runs through the integers 0,1,2,...,n-1,n and α = n -- β then the position vectors
As β run through 1,2,...,n-1, (equivalently α = n - β runs through n-1, n-2, ..., 2, 1) then it is sufficient to show that λ runs through 1/n, 2/n, ..., (n-1)/n in the equation
Intuitively speaking, an array (_,_,_,...) is a collection of blank places to be filled by real numbers. The number that replaces the left most blank is called the first coordinate of the array. The number replacing the next blank to the right is called the second coordinate of the array. If the array has three coordinates, then the number replacing the next blank is called the third coordinate of the array. So for the array (8,9,13) the first, second and third coordinates are 8,9, and 13 respectively.
The size of an array is the number of coordinates in it. The symbols (2,5), (4,--7), (0,0) are examples of arrays of size 2. The symbols (2,5,9), (4,--7,6), (0,0,0) are examples of arrays of size 3. It is obvious how to extend the idea of size to arrays of larger sizes. However, most discussions later will involve arrays of size 2 or 3. Arrays of size 2 are often called ordered pairs. Arrays of size 3 are often called ordered triples. For example, (2,5) and (2,5,9) are an ordered pair and an ordered triple respectively. The customary symbol for an arbitrary array of size 2 is (x,y), and for an arbitrary array of size 3 is (x,y,z). Often the first coordinate is called the x-coordinate, the second coordinate is the y-coordinate and if it exists the third coordinate is the z-coordinate.
Using the term corresponding coordinate(s) it is possible to compare two arrays of the same size. The first coordinates correspond, the second coordinates correspond, and if they exist, the third coordinates correspond. For example, each coordinate in the ordered pair (5,8) is one more than the corresponding coordinate in the ordered pair (4,7) beccause 5 = 4 + 1, 8 = 7 + 1. Later it will be said that (5,8) = (4,7) + (1,1). Each coordinate in the ordered triple (10,14,22) is twice the corresponding coordinate in the ordered triple (5,7,11) because 10 = 2x5, 14 = 2x7, 22 = 2x11. In this situation (10,14,22) = 2(5,7,11). Arrays are useful to indicate the same operation to be done on each number in a collection of numbers.
The statements [1.1], [1.2], [1.3] below are basic rules (axioms) for handling arrays.
[1.1] (Equality of arrays) Arrays are equal if and only if they are of the same size and all the corresponding coordinates are equal.
Notation: (x,y) = (α,β) if and only if x=α and y=β.
Notation: (x,y,z) = (α,β,γ) if and only if x=α and y=β and z=γ.
The arrays (8+2, 7--6) and (10,1) are equal.
The arrays (4--6, 2+3, 7--7) and (--2,5,0) are equal
The arrays (1,2,3) and (1,2,4) are not equal because the corresponding third coordinates are not equal.
The equality of two arrays of size 2 means &means; 2 equations involving equality between corresponding coordinates.
The equality of two arrays of size 3 means &means; 3 equations involving equality between corresponding coordinates.
It is trivial to prove that (x,y) = (y,x) if and only if x = y. This shows that the arrangement of coordinates in an array is important. Therefore, an array of numbers is different from a simple set or collection of numbers.
The familiar addition of numbers may be extended to arrays.
[1.2] (Addition of arrays) Arrays of the same size may be added by adding corresponding coordinates.
Notation: (x,y) + (α,β) = (x + α, y + β)
Notation: (x,y,z) + (α,β,γ) = (x + α, y + β, z + γ)
Examples:
(3,2) + (6,8) = (3+6, 2+8) = (9,10)
(4,3,1) + (5,7,12) = (4+5,3+7,1+12) = (9,10,12).
It is easy to find (x,y) if (x,y) + (4,7) = (0,10). The answer is (--4,3).
Addition can be done with more than two arrays of the same size: simply add the corresponding coordinates of all the arrays:
(x,y) + (α,β) + (τ π) = (x + α + τ, y + β + π)
(x,y,z) + (α,β,γ) + (τ π,ω) = (x + α + τ, y + β + π, z + γ + ω)
The familiar multiplication of numbers may be extended to arrays. However, the most obvious multiplication of corresponding arrays, although simple, turns out not to be very useful. The following multiplication is very useful.
[1.3] (Product of a number and an array) An array may be multiplied by a number, by multiplying each coordinate in the array by that number.
Notation: λ(x,y) = (λx,λy);
Notation: λ(x,y,z) = (λx,λy,λz);
2(3,4) = (2x3,2x4) = (6,8).
3(5,6,7) = (3x5,3x6,3x7) = 15,18,21).
Notice that the same results can be obtained by repeated addition since the multiplying factors 2 and 3 are natural numbers:
2(3,4) = (3,4) + (3,4) = (6,8)
3(5,6,7) = (5,6,7) + (5,6,7) + (5,6,7) = (15,18,21).
[1.4] (Addition is commutative) Two arrays of the same size may be added in either order.
Notation: (x,y) + (α,β) = (α,β) + (x,y)
Notation: (x,y,z) + (α,β,γ) = (α,β,γ) + (x,y,z).
The proof is based on the commutativity of real numbers, x + α = α + x, y + β = β + y,
z + γ = γ + z (for arrays size 3)
[1.5] (Addition is associative) For the sum of three arrays of the same size, either addition may be done first.
Notation: ((x,y) + (α,β)) + (σ,μ) = (x,y) + ((α,β) + (σ,μ))
Notation: ((x,y,z) + (α,β,γ)) + (σ,μ,ω) = (x,y,z) + ((α,β,γ) + (σ,μ,ω)).
There are two additions to be done, indicated by + and +. On the left side of the equation the red addition + is done first, then the second addition is done +. On the left side of the equation the black addition + is done first, and then the red addition + is done.
Proof is based on the associativity of real numbers and the equality of arrays:
(x + α) + σ = x + (α + σ),
(y + β) + μ = y + (β + μ),
(z + γ) + ω = z + (γ + ω) (for arrays size 3)
[1.6] (Additive identity) The array with all coordinates equal to zero is the additive identity.
Notation: (0,0) is the additive identity for ordered pairs because (x,y) + (0,0) = (x,y).
Notation: (0,0,0) is the additive identity for ordered triples because (x,y,z) + (0,0,0) = (x,y,z).
Proof is based on the fact that 0 is the additive identity for real numbers: x + 0 = x ... etc.
[1.7] (Additive inverse) Every array has an additive inverse obtained by negating all of the coordinates.
Notation: (--x, --y) is the additive inverse of (x,y). --(x,y) will denote the additive inverse of (x,y).
Notation: (--x, --y, --z) is the additive inverse of (x,y,z). --(x,y,z) will denote the additive inverse of (x,y,z).
Simple computations show that (x,y) + (--x, --y) = (0,0) and that (x,y,z) + (--x, --y, --z) = (0,0,0).
[1.8] (Subtraction) One array may be subtracted from another array of the same size by subtracting the corresponding coordinates in that order.
Notation: (x,y) -- (α,β) = (x -- α, y -- β)
Notation: (x,y,z) -- (α,β,γ) = (x -- α, y -- β, z -- γ)
[1.9] (Left distributive) The product of a real number and the sum of two arrays of the same size is equal to the sum of the products of that real number and each of the arrays.
Notation: λ[(x,y) + (α,β)] = λ(x,y) + λ(α,β)
Notation: λ[(x,y,z) + (α,β,γ)] = λ(x,y,z) + λ(α,β,γ)
[1.20] (Right distributive) The product of the sum of two real numbers and an array is equal to the sum of the products of each real numbers and the array.
Notation: (λ + σ)(x,y) = λ(x,y) + σ(x,y)
Notation: (λ + σ)(x,y,z) = λ(x,y,z) + σ(x,y,z)
There are statements that are similar to those from [1.1] to [1.20] that apply to arrays of larger sizes.
From graphs in elementary algebra, the mid point formulas for array3 are:
Multiplying all expressions by 2 produces the desired equation (#) above for plane and space.
Recall that O is the name of the origin, and it is located by the zero array = (0,0) or the <zero array = (0,0,0).
Construct the addition parallelogram for the vector sum p + q which is shown in the adjacent figure as parallelogram OPSQ. Diagonal vector OS = p + q. Assign array1
to point P and array2 to point Q. Assign array4 to point S. Needed to show is
A theorem in plane geometry says that the diagonals of a parallelogram bisect each other. In the figure, M is the midpoint on both diagonals OS and PQ. Therefore, by the discussion about midpoints above, both the sums