7A. Addition of Vectors multiples of a Vector k - Coordinate viewpoint
Signed Number Multiples of VectorsAddition: August 11, 2008: Along a line, a displacement u has a length and a direction. It may be depicted by an arrow.
It may be called a vector. The displacement may be doubled: 2u: |=====>|=====> 2u: |============> or multiplied by any proper or improper
fraction. The direction of the displacement u may also be reversed, and the result denoted by -u: -u: <===== We take
and so on. Exercise: Draw -1.5u, -0.5 u, -1¼ u, -0.75u |
Spanning or Measurement Assumption
Description or measurement of direction and length for directed line segments using a unit length or arrow, and numbers prefixed with a negative sign.
Vector Measurement Assumption (ii): For each horizontal non-zero vector k and each horizontal vector v, that is a unique real number m > 0 such that v = m k or v = -m k, but not both unless m = 0. We further assume a vector v and all its horizontal translations have the same length m (are the same multiple m of the length of k.)
The positive sign + is optional in front of unsigned numbers.
Length Measurement Assumption (i): For each non-zero length k and each length v, there is unique real number m > 0 such that v = m k.
The measurement assumption can be illustrated with the use of a ruler or tape measure with unit length given from metric or imperial system: 1cm, 1 inche, 1meter, 1 km, etc. In this, the choice of the unit length k and/or unit vector k is arbitrary.
Further Assumption: The length and direction of a directed line segment (vector) is determined by its coordinates (length and direction) with respect to a unit vector.
Instructional Note - Expositional Options to consider: We could write the vector before the coefficient. Then 1.9 k would be written as k(1.9) and -1.7k would become k(-1.7) and (-k)(1.7) The option would put k besides the positive and negative signs used as prefixes and k itself could be viewed as another prefix. That might aid the exposition below. Another option might be to write (5, 0 degrees)k and (5, 180 degrees)k for 5k and -5k respectively for consistency with and as an aid to a later introduction of complex numbers.
7B. Length and Directions of Vector Sums
When collinear vectors with the same direction are added, the length of resulting vector (resultant) is another vector in the same direction and with length given by the sum of the lengths of the addends.
a units b units
(o)=======>=============>
(o)===================>
a + b units
When collinear vectors (displacements) with opposite direction are added, the length of the resultant vector is zero in the case the addends are additive inverses:
a units
(o)===========>
(o)<===========
a units
In the latter case the result has length
0 & it vanishes.
When collinear vectors (displacements) with opposite direction are added,
the length of the resultant vector is the difference a - b units where the
smaller length, say b units, is subtracted from the larger length a units, and
the direction of the longer gives the direction of the resultant.
a units
(o)========================>
(o)============><===========
(a - b) units b units
7C. Addition of Multiples of a Unit Vector k
Let k be a unit vector
k
===>
Then we may define real number multiples of k
4.5k : =================>
2k : ========>
-3k : <===========
-2.5k : <=========
Relative to the length of k, our unit length, these vectors or displacements have length 4.5, 2, 3 and 2.5 units.
Addition of Negative Multiples
vectors in the same direction
The sum of the two negative multiples
-3k : <===========
-2.5k : <=========
is calculated as follows
3 units 2.5 units
<===========<=========(o)
<=====================(o)
(3 + 2.5) units or 6.5 units)
The sum is thus (3+2.5)(-k) = -5.5 k, or
(-3)k + (-2.5)k = (-5.5) k.
The direction is another negative multiple. The addends and the resultant all have the same direction. Observe how we add the lengths and how the negative sign is kept.
The result = (longest length + shortest length) (direction of BOTH
Addition of Positive Multiples
vectors in the same direction
The sum of the two positive multiples
4.5k : =================>
2k : ========>
is calculated as follows
4.5 units 2 units
(o)=================>========>
(o)==========================>
(4.5 + 2) units (or 6.5 units)
The sum is thus (4.5+2) k = 6.5 k, or
(+4.5)k + (+2)k = (+6.5)k.
The direction is another positive multiple. The addends and the resultant all have the same direction. Observe how we add the lengths and how the positive sign is kept.
The result = (longest length + shortest length) (direction of BOTH
Addition of Positive and Negative Multiples
vectors in opposite direction with
positive multiple shorter than negative multiple
The sum of the positive and negative multiples
2k : ========>
-3k : <===========
is calculated as follows
3 units
<===========(o)
========><==(o)
2 units (3-2) units
The sum is thus (3-2)(-k) = -(3-2) k, or
(+2)k + (-3)k = (3-2)(-k) = -k.
Conclusion:
The result = (longest - shortest length)(direction of longest)
In this, the direction of the longest multiple (the negative one) gives the direction of the sum
Addition of Positive and Negative Multiples
vectors in opposite direction with
positive multiple longer than negative multiple
The sum of the positive and negative multiples
4.5k : =================>
-2.5k : <=========
is calculated as follows
4.5 units
(o)=================>
(o)=======><=========
(4.5 - 2.5) units 2.5 units)
The sum is thus (4.5-2) k = 2 k, or
(+4.5)k + (-2.5)k = (4.5-2.5)k = 2k.
Conclusion:
The result = (longest - shortest length)(direction of longest)
In this, the direction of the longest multiple (the positive one) gives the direction of the sum
Exercises:
Express the following as a multiple of k
- z = 8k + 9k
- y = 4k + -6k
- x = -8k + - 7k
- w = 5.5 k + (-3.5) k
The foregoing suggest how to add real numbers in a manner that multiplication by sums of signed real numbers distributes over collinear vector addition
(A+B)k = Ak + Bk
See the next lesson on addition of unsigned numbers for proof.
Number Theory & Practices
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