21. Addition of Multiples of a Single Unit Vector k
Let k be a nonzero vector. It lengths provides a unit of measure.
So we take k be our unit vector -instead of the letter u employed in the previous
lessons. That is an accidental change of notation.
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.
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