What is the answer to this problem?
1 answer:
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Answer:
From top to bottom, the boxes shown are number 3, 5, 6, 2, 4, 1 when put in ascending order.
Step-by-step explanation:
It is convenient to let a calculator or spreadsheet tell you the magnitude of the sum. For a problem such as this, it is even more convenient to let the calculator give you all the answers at once.
The TI-84 image shows the calculation for a list of vectors being added to 4∠60°. The magnitudes of the sums (rounded to 2 decimal places—enough accuracy to put them in order) are ...
... ║4∠60° + 3∠120°║≈6.08
... ║4∠60° + 4.5∠135°║≈6.75
... ║4∠60° + 4∠45°║≈7.93
... ║4∠60° + 6∠210°║≈3.23
... ║4∠60° + 5∠330°║≈6.40
... ║4∠60° + 7∠240°║≈ 3
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In the calculator working, the variable D has the value π/180. It converts degrees to radians so the calculation will work properly. The abs( ) function gives the magnitude of a complex number.
On this calculator, it is convenient to treat vectors as complex numbers. Other calculators can deal with vectors directly
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<em>Doing it by hand</em>
Perhaps the most straigtforward way to add vectors is to convert them to a representation in rectangular coordinates. For some magnitude M and angle A, the rectangular coordinates are (M·cos(A), M·sin(A)). For this problem, you would convert each of the vectors in the boxes to rectangular coordinates, and add the rectangular coordinates of vector t.
For example, the first vector would be ...
3∠120° ⇒(3·cos(120°), 3·sin(120°)) ≈ (-1.500, 2.598)
Adding this to 4∠60° ⇒ (4·cos(60°), 4°sin(60°)) ≈ (2.000, 3.464) gives
... 3∠120° + 4∠60° ≈ (0.5, 6.062)
The magnitude of this is given by the Pythagorean theorem:
... M = √(0.5² +6.062²) ≈ 6.08
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<em>Using the law of cosines</em>
The law of cosines can also be used to find the magnitude of the sum. When using this method, it is often helpful to draw a diagram to help you find the angle between the vectors.
When 3∠120° is added to the end of 4∠60°, the angle between them is 120°. Then the law of cosines tells you the magnitude of the sum is ...
... M² = 4² + 3² -2·4·3·cos(120°) = 25-24·cos(120°) = 37
... M = √37 ≈ 6.08 . . . . as in the other calculations.
Proof by contadiction.
Let assume there exist such positive integers and
whose sum is a prime number
, that their greatest common divisor is greater than 1.
Since is a prime number and
, then
, but we assumed earlier that
, and there are no two positive integers that sum up to 1.
q.e.d.