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MCV4U - Unit 6 ~Vectors~, Vectors, Common Vector Applications,…
MCV4U - Unit 6
~Vectors~
Vectors
In this course, they are either in:
R^2
2d
Has 2 dimensions, x and y
unit vectors: i = (1, 0) and j = (0, 1)
vector (a, b) is ai + bj
There is an x-axis (y = 0) and a y-axis (x = 0)
R^3
3d
Has 3 dimensions, x, y and z
Think of z as the height
unit vectors: i = (1, 0, 0),
j = (0, 1, 0), k = (0, 0, 1)
vector (x, y, z) is xi + yj + zk
There is an yz-plane
(x = 0), xz-plane (y = 0), and xy-plane (z = 0)
HAVE
Magnitude
In R^3: (a^2 + b^2 + c^2)^1/2
In R^2: (a^2 + b^2)^1/2
Essentially the length of the vector
Always positive
Written as: |vector| (absolute value)
Can be scaled up and down
ex: v = (a, b)
2v = (2a, 2b)
-v = (-a, -b)
This is actually the NEGATIVE/OPPOSITE vector, which means it has the same magnitude, but opposite direction
ex: ---> and <---
HAVE
Direction
(x2 - x1, y2 - y1)
Usually given as NWSE with an angle associated
ex: (N 30degrees W) would mean to start at North, then go 30 degrees to the westerly direction
Don't have a set location, and can therefore be moved around
So vectors with the same magnitude AND direction are equivalent. Vectors with the same magnitude and OPPOSITE directions are negative pairs/opposite vectors
Are written with a (-->) on top (but that wasn't possible to do in this conceptmap)
Common Vector Applications
Velocity vectors/moving objects
ex: An airplane is flying at a speed of 150km/hr heading in the direction N 45degrees E. What is the vector?
(150cos45, 150sin45)
Trigonometry + cosine law
ex: If the magnitude of a is 3, and the magnitude of b is 2, and the angle between then is 60degrees, what is the magnitude of a + b?
Use cosine law to solve for c, which in this case is the magnitude of a + b :
|a + b|^2 = |a|^2 + |b|^2 - 2(|a|)(|b|)cos60
= (19)^1/2
Operations with Vectors
Combining Vectors
Algebraically
If v1 = m(x, y) and v2 = n(a, b).
v1 + v2 = (mx + na, my + nb)
v1 - v2 = (mx - na, my - nb)
Graphically
A --> B + B --> C
= A --> B
Forms a triangle
A --> B + A --> C
= A --> D
Which is the diagonal of the parallelogram
A --> B - A --> C
= A --> B + C --> A
When subtracting, reverse the vector (make it negative) (same magnitude but opposite direction)
Component Factors
As mentioned, you can scale up and scale down vectors
Therefore a vector can be expressed as a scalar combination of the other 2 vectors IF THEY ARE ON THE SAME PLANE/AXIS
ex: v1 = (x, y)
v2 = (a, b)
v3 = (c, d)
v1 expressed in terms of v2 and v3:
Let v1 = t(v2) + s(v3)
so: x = ta + sc
y = tb + sd