Week 4 Flashcards
Can vectors be added?
Yes, if they both have elements of R^d then they can be added, component wise.
Can vectors be multiplied?
Yes, an element of R^d can be multiplied by a scalar c in all real numbers
How can vectors be representd by an arrow?
A 2-dimensional vector (a b) can be represented by an arrow from the origin (0, 0) to (a, b).
What is a vectors?
In general, a vectors describes displacement therefore it has no defines displacement from one point to another.
When adding vectors u and v, how are they geometrically represented?
- u stays the same
- v is moved to have a starting point at the end point of u
- Create the line v from this starting point
- Generate the line u+v by connecting the statying point of u to the end point of v.
NOTE: These vectors DO NOT need to start from the origin
Do vectors have a fixed position? What are their fixed properties?
Yes
However, they have a direction and a length (magnitutde)
When multiplying vectors u and v, how are they geometrically represented?
- New vector is generated which is c times longer than u
2. If negative c, direction reverses
Can vectors be subtracted?
Yes, remember that a-b = a + -(1)*b
Therefore u - b = u + (-1)*b
When subtracting vectors u and v, how are they geometrically represented?
- u stays the same
- v is reversed then v is moved to have a starting point at the end point of u
- Create the line v from this starting point
- Generate the line u+v by connecting the statying point of u to the end point of v.
NOTE: These vectors DO NOT need to start from the origin
How to determine line interval joining points u and v?
au + (1-a)v for a in [0, 1]
What is median of a triangle?
Take a line from a corner point to the middle of the opposite.
Thus, there are 3 medians in a triangle
What does th dot product tell us?
If dot product = 0, orthogonal
Two vectors in 2 or 3 dimensionals are perpendicular iff they are orthogonal.
(angle between them is 90 degrees)
ALSO
If they are pairwise orthogonal, thay are linearly independent
Are linearly independent vectors always orthogonal?
No
What information do integrals provide?
They give us the TOTAL CHANGE to a functions value in an interval for which the RATE OF CHANGE is KNOWN.
i.e.
use intergrals to determine distannce travelled in period time IF we know speed at any given time
use integrals to derive area or volume of complicated objcts
What is the definite integral?
This is the area of a function f between two points a,b. The area bound bt f, the x axis and the verticals lines at x = a and x = b is the signed area.
