Head to Tail and Parallelogram methods follow for vector addition. Can you see how the operations on coordinate justify these geometric (coordinate-free) methods?
1. Head-To-Tail Method
A straight line arrow from one point to another may summarize the movement of an object. The object itself may follow a curved path between the tail or initial point of the arrow and the head or terminal point. Similarly when a sequence of straight line motions is followed, one after another, the arrow joining the initial point of the first motion to the terminal point of the last motion summarizes or gives the sum or resultant of the intermediate motions. That is a context and motivation for the head to tail addition of a sequence of arrows or vectors in navigation.
When two movements or motions, one after another, are summarized by arrows with the head of the first, its terminal point, at the tail of the second, the second's initial point, the sum or resultant of these two movements, vectors or arrows is summarized by the the arrow joining the initial point of the first motion to the terminal point of the second. This describes the head-to-tail method of adding two arrows or vectors together. The head of the first must be at the tail of the second. Here is a context and motivation for the head to tail addition of a sequence of arrows or vectors in navigation, a special case of the previous lesson.
Head to Tail Addition Method
2. Parallelogram Method
When two arrows or vectors (representing motions if you wish) have a tail at the same place, they may be added together by moving the tail of one to the head of the other with the aid of a parallelogram, and then using the head to tail method for addition. This gives the parallelogram method for adding a pair of arrows or vector addition. The resultant arrow does not depend on which arrow, the first or second, is moved. Here is a repeatable and reproducible methods, arbitrarily defined, for vector or arrow addition. More generally, parallelograms can be used to displace or move arrows from one location to another without changing their lengths or directions. But that is another story, or chain of reason, not illustrated here.
The following diagram shows in four steps 1, 2, 3 and 4 how to add two vectors which start at the same place, or have the same origin.
Step 5 just gives the shorthand notation for a vector from a point A to B in a figure. Outside of a figure, I will use the notation AB to denote the same vector because it not easy to put an arrow over the two letters AB. The parallelogram law says how to compute AB + AC.
Saying how to add vectors or "straight-line movements" which have tails or starting points defines a second kind of vector addition. The first first kind, head to tail addition of vectors representing movements, was described in the previous lessons. In mathematics, you should remember saying how to do a calculation or operation defines it. Remember this principle. Vector addition represents as given here represents a geometric calculation.
From the parallelogram rule or law for addition, the calculation of the sum of vectors AB + AC and the calculation of AC + AB is identical. So the order of addition is not important. Expressions of the form AB + AC can be replaced by the expression AC + AB when we describe calculations with vectors (arrows). That is, vector addition commutes. Remember this when we talk about real and complex numbers below.
The applet illustrates the parallelogram law for the addition of vectors with tails at the origin.
3. Component Method
In a plane, the intersection of two perpendicular lines, one horizontal and the other vertical, defines a reference point or origin for the plane. Each arrow in the plane is equal to the sum or resultant of a horizontal and vertical arrows, its so-called horizontal and vertical components. This representation or decomposition of an arrow as the sum of horizontal and vertical components leads to a third method for arrow addition given by the addition of components. The horizontal components of an arrow sum is given the arrow sum of the horizontal components. Likewise, the vertical components of an arrow sum is given the arrow sum of the vertical components. Here is a technical observation with little motivation except for consequences that will follow.
Each vector from the origin in a rectangular coordinate is the sum of vectors parallel to the coordinate axis's. The following diagram shows how these "component vectors" parallel to the axis's may be used instead of the parallelogram rule.
See applet for examples. Have it display rectangular coordinates for points and vectors. To learn more, visit the chapter vectors in Calculus and Beyond.
Coordinate View (Again)
If [a,b] and [c,d] are the heads of two vectors, we the head of the sum of the vectors will be at location [e, f] = [a,b] +[c,d] where the + operation indicates arrow addition. In the case where both heads are in the first quadrant, we have e = a +c and f = b +d. In the case where one or both are not in the first quadrant, put a + c = e and b + d = f. This defines the addition of coordinates. Exercise: Show this addition is well-defined.
Observe, addition of vectors using the first method (parallelogram method) commutes. That the order of addition does not affect the result. Therefore addition commutes in all further methods that produce the same result. This implies the addition of coordinates just defined commutes
4. Coordinate Method
In a plane, the intersection of two perpendicular lines, one horizontal and the other vertical, defines a reference point or origin for the plane. The head of each arrow in the plane has coordinates. Ordered pairs of vertical and horizontal coordinates, ordinates and abscissa, can be employed to add arrows together or find the position of the head of their sum, when the tails of the vectors in the sum are both located at the origin. This gives a fourth method for arrow addition given by the addition of coordinates. This method is very similar to the third method for addition with components. Here is another technical observation with little motivation except for consequences that will follow.
The next figure shows each vector as a sum of components. We assume the vector heads are located at [a,b] and [c,d]. Each vector from the origin may be defined (or drawn) by giving the location of its head.
The next step is to arrive at a formula for vector addition in terms of the coordinates [a,b] and [c,d] of the head locations of the vectors or summands in this vector addition.
The foregoing suggests we represent points in the plane and vectors ending at those points by the coordinates of the head of the vector, that is the coordinates of the points. The foregoing also suggests the vector sum of the arrows with heads at [a,b] and [c,d] respectively has its head at [a+c, b+d]. So to represent the vector addition of arrows, we put
[a+c, b+d] = [a,b] + [c,d]
This gives the coordinate way to add points and their position vectors in the plane. The position vector of a point goes from the origin (its tail) to the point (its head).
See applet for examples. Have it display rectangular coordinates for points and vectors.
5. Recap and Examples
The sum of two points with the rectangular coordinates [a,b] and [c,d] is given by [a+c,b+d]. We therefore write
[a,b]+[c,d]= [a+c,b+d].
The addition rule is simple add the rectangular coordinates to get the rectangular coordinates of the sum. This coordinate addition follows from our previous discussion of methods for arrow addition.
Example [2,5]+ [6,2]=[8,7].
| 7 !=======+------------//-------* [8,7] | % | [2,5] % 5 |=======* | | | | | | | | | [6,2] | 2 |-------|-//----------* | | | % | | | % | ------------------------------------------------- | 2 6 8 | |
Figure 2. Addition of Points [2,5] + [6,2] = [6+2, 5+2]
Scaling Distributes
Scalar Multiplication and its Distributive Law. The repeated addition of an arrow to itself, n-1 additions, leads to the notion of a scalar multiple: n times the arrow. Drawing parallelograms, tessellating the plane with them, implies or suggests that multiplication of vectors by whole numbers and then fractions distributes over the sum of two different vectors. Here is motivation if not a proof, for the first distributive law, namely the distributive of scalar multiplication over vector addition. Here is another technical observation with little motivation except for consequences that will follow. Note: the forthcoming geometry will give an alternate viewpoint or a derivation of the distributive law based on notions and assumptions about similarities of triangles.
Multiplication by Unsigned Numbers (Scaling)
Here 2A denotes A + A while nA = A added to itself, n-1 times. See A, 2A, 3A and 4A in the diagram below.
Obtaining the Distributive Law for Scaling
The following diagram also suggests that n(B+C) = nB + nC whenever n is a whole number 0, 1, 2, 3, 4, ....
Similar reasoning with B and C replace by (B/m) and (C/m) suggests
(n/m) (B +C) = (n/m) B + (n/m) C
So multiplication by a fraction distributes over addition. Beyond this calculations with real numbers > 0 are done by approximation by decimal fractions, improper or not.
Geometric arguments using distances in the plane and appeals to error control in approximations (or continuity) imply
r (B+C) = rB + rC
whenever r > 0 is a nonnegative real number with B and C position vectors in the plane (initial point at the origin).
With the foregoing motivation, we make two assumptions.
Assumption 1: Distribution Law for Multiplication of Vectors by unsigned numbers:
r (B+C) = rB + rC
whenever r > 0 is a nonnegative real number with B and C position vectors in the plane (initial point at the origin).
Assumption 2: Distribution Law for Multiplication of Unsigned Numbers by unsigned numbers:
r (b+c) = rb + rc
whenever r, b and c are unsigned numbers > 0
Both assumptions will be used in the further chain of reasoning below.
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