3.1: Vectors and Scalars #

3.2: Addition of Vectors - Graphical Methods #

when considering displacement on a line, addition (and subtractino) is easy
displacement when someone walks 10 km east, then walks 5 km (i.e a quantity with both value and direction), change in position = ?
given the above example, one could graph this and see the following »>
because the vectors are perpindicular, we can use the pythagorean theorem to get the resultant vector
the max of a resultant vector can only be equal to the sum of the 2 vectors (their magnitudes) (as if they were on a straight line)
- consider why all 90 degree lines that are added to make a vector can only produce perpindicularity
formal graphical definition of vector addition “the arrow drawn from the tail of the first vector to the tip of the second vector is the sum of the two vectors”
there are various methods to add vectors visually » including tail to tip and the parallelogram method, both ends of resultant vector cannot be touching the tips of component vectors (for addition)

defintion is intutive, but the negative of a vector is simply the same magnitude but opposite in direction (recall magnitude cannot be negative)
to visually subtract vectors, consider the vector equation V2 - V1 # add this in later
how would we get the difference of the two vectors

components of a vector are usually in perpindicular directinos (such as the x and y axse)
given the resulation of a velocity vector, we can use trig to solve for the components
thus, we have 2 ways to specify vectors: 1. by components (Vx and Vy) 2.by magnitude V and direction theta it makes with positive x axis
the following example illustrates non-perpindicular compponents »
is theta the angle v makes with the positive x axis

Draw a diagram, adding the vectors (graphically) - either by parallelogram method or tip to tail

do the rest after doing the homework

a unit vector is defined to have a magnitude exactly equal to one in an x,y,z coordinate system they are called // hats
all vectors are simply scaled versions of the unit vectors (i hat, j hat, k hat) (x,y,z)
V = Vxi + Vyj + Vzk // add math later

// instantaneous def of veloicty and acceleration
in 1D, displacement was defined as a change in position
however, in the case of 2 or 3 dimensions, the displacement vector is the vector representing change in position
// eq.s
instantaneous velocity is the limit of the average velocity as change in time approaches zero
- direction of the instantaneous veloicty is along the line tangent to the path at that moment
instantaneous acceleration is defined as the limit of the average acceleration vector as the time interval change in t is allowed to approach 0 instantaneous acceleration will be non-zero when the magnitude of velocity changes or if the direction changes
in 2D or 3D, constant acceleratino means ax, ay, az = 0 and the average acceleration is = instantaneous acceleration at any moment

path of projectile motion is parabolic
currently we are only discussing when the only acceleration is downward (gravity) - we are ignoring air reistance
basic info to solve problems
- g = 9.80
- t = 0
- x0 = y0 = 0

// revisit once done with hw

Read carefully (choose the object to analyze)
Draw a careful diagram showing what is happening to the object
Choose an origin and an xy coordinate system
Decide on the time interval, for which the object can only be under the motion of gravity x and y motions are conneceted by the common time
Examine x and y components seperately → if given intial velocity, resolve into x and y components
List known and unknown quantities, using ax = 0, ay = -g / +g, and + / - dephending on whether you choose y positive down or up
helpful tips #
- Vx never changes
- Vy = 0 at highest point of any trajectory that returns downward
- velocity just before landing is generally not zero
Think/plan before eq.s → apply relevant eqs and combine eqs if neccesary

If 2 particles are on the same line, determing their velocities, relative to each other, is a matter of simple addition / subtraction
When they are not along the same line, vector addition is used.
Label carefully to maintain distinct and unambigious reference frames subscripts