Physics Notes, Chapter 3: Two Dimensional Motion and Vectors
I. Introduction to Vectors
A. Scalars and Vectors
1. Scalar quantities, such as speed, use only magnitude while vector
quantities, such as velocity, use magnitude and direction
2. In the text, scalar quantities are in italics, vector quantities are bolded
3. In drawings, vectors are shown by arrows, the length of which indicates its
magnitude, the direction its pointing indicates the direction
4. The resultant is the answer that results when two or more vectors are
added together, or one vector has the same net effect as two or more
vectors
B. Properties of Vectors
1. Vectors can be aligned in a drawing so that the tip of one is right at the
beginning of the next, so it becomes easy to draw in the resultant vector
from the start of the first vector to the tip of the last
2. Triangle method of addition--graphically or mathematically determining
resultant vectors by moving vectors parallel to themselves in a diagram to
align them
3. Vectors can be added in any order
4. When subtracting vectors, add the opposite (+/- or west/east, for example)
5. When multiplying or dividing vectors by scalars, the result is a vector
II. Vector Operations
A. Coordinate Systems in two dimensions
1. The x and y axis can both be used at the same time to simplify the
direction a moving object is traveling
2. X is usually east, or along the horizon (whichever the problem calls for),
while Y is usually north, or vertical
B. Determining Resultant Magnitude and Direction
1. The magnitude of a resultant of two perpendicular vectors can be found
using the Pythagorean theorem
2. The direction of the resultant can be found using the tangent function (or
sine, or cosine)
C. Resolving Vectors into Components
1. Components are the horizontal and vertical parts that can be added to
describe a vector
2. Resolving (or breaking) a vector into two components helps to find the
unknowns in a problem
3. Again, the Pythagorean theorem and simple trig functions are used in
calculations
D. Adding Vectors That Are Not Perpendicular
 This can be done by resolving each of the vectors involved into their x and
y components, and then adding x components together, and adding y
components together, and then finally applying the Pythagorean theorem
and trig functions
III. Projectile Motion
A. Projectile-objects that are thrown or launched into the air are subject to gravity
B. Projectiles have an initial horizontal velocity and follow a parabolic path
when air resistance is neglected (called a trajectory)
1. Note that a projectile and an object simply dropped, if both released at the
same time and if air resistance is ignored, will fall at the same rate and hit
the ground at the same time
2. When the motion of two objects described above is examined by resolving
their velocities into a horizontal and vertical component, then this is easy
to explain (pg. 99-100)
C. To find the velocity of a projectile at any point during its flight, find the vector
sum of the components of the velocity at that point—use the Pythagorean
theorem to find the magnitude of the velocity, and simple trig functions
(tangent) to find the direction
D. Projectiles launched at an angle to the horizontal
1. Again, motion of the projectile, this time its initial velocity, must be
analyzed by resolving it into its components
2. Some more or less useful equations for projectiles launched at an angle:
x = vi (cos ) t
vx = vi (cos ) = constant
y = vi (sin ) t – ½ g (t)2
vy,f = vi (sin ) – g t
vy,f2 = vi2 (sin )2 – 2 g y
IV. Relative Motion
A. Frames of Reference—velocity measurements differ in different frames of
reference, so it is often important to define where the observer is in problemsolving
B. Relative Velocity—approach these problems by clearly defining quantities
(such as velocities) with subscripts that explain both what object the given
refers to, and with respect to what frame of reference (i.e. the Earth, the other
car…)