Physics Ch.2A- Vectors
Vector addition: A+B (3x+3y)+(2x-1y)=?
(3x+2x)+(3y-1y)=5x+2y
Dot product: A*C (3x+3y)*(-1x)=?
3x(-1x)+3y(-1x)=-3xx-3yx=-3(1)-3(0)= -3
Is a mathematical LINE a real being or a being of reason? What is the concrete ORIGIN of the concept? →
A BEING OF REASON. It originates from the mental construct of mathematical SPACE, where we think of something that has extension but no actual parts and is potentially divided in any way we want.
Is a completed infinity a real being or a being of reason? ∞
A being of reason.
Is "nothing" a real being or a being of reason? Can nothing exist outside of the mind?
A being of reason. No, all that exists is something.
What is the basic concept of a vector?
A line of certain LENGTH with a DIRECTION.
What is the formal definition of vector? (p.42)
A mathematical entity of a given MAGNITUDE and DIRECTION, where magnitude and direction are taken in many varied ways so that the concept is APPLICABLE to many QUALITATIVELY different situations.
What is abstraction?
A process of the mind by which I LEAVE OUT certain aspects of reality.
How is a vector symbolized?
A symbol with a ray over it. (ex: A⁻ )
What is the POSITION vector? What is the significance of the DIFFERENCE of two position vectors?
A vector with a TAIL at the origin and head at the point of interest (position). The difference between two position vectors tells us HOW FAR and in what DIRECTION one must go to get from position 1 to position 2.
What is the distinction between ACTUAL and POTENTIAL? How does that distinction apply to a mathematical LINE?
Actual refers to what is. Potential refers to what can be. A mathematical line has no actual parts but is potentially divided in any way we want.
Why are the vectors illustrated in figure 2-4 examples of MIXED TYPE (quantity and quality)?
Because BRIGHTNESS is a quality, but the measure of its INTENSITY per unit area involves the area of a square, which itself can be thought of as composed of squares of a given length- (length)² and so AREA is an analogous way measured by length, which is quantity.
What are the 2 basic categories of vectors? (w/ examples)
PURELY QUANTITATIVE; it is something that literally has length and direction (ex: meeting of two walls) ANALOGOUS USE; where vector means the magnitude and direction of a quality (ex: the intensity of brightness)
Dot product
Parallel product, commutative (non-directional)
Cross product
Perpendicular product
What type of vector is a DISPLACEMENT---quantity, quality, or a combo?
Quantity
What is the subject of mathematics? Where does it come from?
Quantity, It comes from the REAL WORLD.
Who gets credit for the Cartesian coordinate system? Who is known to have used rectangular coordinates before him?
Rene Descartes (xyz), Nicole Oresme (xy)
What 2 meanings does length have?
The EXTENSION of a line The MEASURE of that extension
What is a vector field? ↑↑↑↑
The assignation of a vector to EVERY POINT in space, or every region in a space.
Which of the Oxford Calculators may have been the first to describe MOTION?
Thomas Bradwardine
What is ANALOGY? Why does analogy always point to some primary example (called PRIME ANALOGATE)?
Use that is like in SOME WAYS but not in others. Prime analogate is the PRIMARY use.
Can you conceive of a substance leaving out its properties (time, its environment, its orientation, or its place, its relation to other things, without color, without extension)?
Yes to all, except EXTENSION. Without at least extension there is NOTHING that can be conceived.
Angle of dot product
cosø=(A*B) / (|A||B|)
Angle of cross product
sinø=(|AxB|) / (|A||B|)
Cross Product: AxB (3x+3y)x(2x-1y)=?
x(0)-y(0)+z(-9)= -9z
Magnitude (formula)
|A|=√((Ax^2)+(Ay^2)+(Az^2))
Unit vector (formula)
Â=A/|A|
How is a VELOCITY vector an ANALOGOUS use of vector? What is the PRIMARY MEANING of vector that this refers to?
It is analogous on two distinct levels: (1) it treats TIME as if it were a quantity (2) the use of a RATIO (length to time) It refers to magnitude (speed) and a direction of locomotion.
When referring to two vectors, what does DEGENERATE mean?
It means they are PARALLEL to each other. ‖
Is quality essentially reducible to quantity?
No
WHAT is the mean speed theorem, and WHO first formulated it?
For a uniformly accelerated body which begins from REST, the constant speed (the mean speed) that one would have to travel in order to traverse the SAME DISTANCE in a given time as the uniformly accelerated body is given by: Vmean= (Vf)/2, where Vf is the final velocity of the accelerated body. The Calculators were the FIRST to formulate this theorem.
A ray pointing "up" leaves out what aspect of up? ↑
Gravity
Where does mathematics originate?
Mathematics ultimately originates in SENSE KNOWLEDGE. It is about the REAL WORLD.
WHO were the Oxford Calculators and WHEN did they live?
Most notably: Thomas Bradwardine (c.1290-1349) William Heytesbury Richard Swineshead (c.1340-1354) John Dumbleton
Who first determined the law of uniform acceleration? How long was this before Galileo?
Nicole Oresme (almost 300 before)
