• Recall that “vectors” are arrows that
represent a vector quantity (magnitude and
– The length of the arrow represents the
magnitude of a measurement
– The direction of the arrow within a coordinate
system represents the direction of the vector.
• Most vector directions are references to a particular
direction, i.e., 0o or say the “west” or negative x-axis
of a graph.
• Vector quantities of the same units can be added
to each other.
• Vectors can be added in 2 ways:
• To add 2 vectors geometrically simply place the
tail of one vector to the head of the other. The
solution, or Resultant vector, is the vector from the
point of origin of the 1st vector to the end point of
• When adding vectors geometrically, one often
places the first vector at the origin.
R = magnitude R
@ o North
Adding Vectors Algebraically
• Adding vectors algebraically involves
simply adding magnitudes that are along the
The Unit Vector
• A unit vector is a vector of 1 “unit” in
length that defines a particular direction.
– For example: In a Cartesian system, there are 3
principal axes: x, y, & z. The unit vector of
each is simply a vector of length 1, in each
• Often a vector may be written as the sum of
its parts, each multiplied by a unit vector
giving the direction associated with that
The vector R shown at the right
R could be written as R = 3x + 4y
x Notice that the vector R can be
3 Represented as the geometric sum of
3 times the x-unit vector plus 4 times
The y-unit vector.
• i, j, k notation
• The resultant vector obtained from the
graphical addition of two vectors can be
found form adding the vectors algebraically.
1st: Break each vector into independent
components. I.e., Put the vector into i,j,k
notation if it’s not already.
2nd: Add each component of both vectors
independently of the other components.
– A = 3i + 4j +5k
– B = 2i +7k
R = 3i + 4j +5k + 2i +7k OR 3i + 4j +5k
+ 2i + 0j +7k
R = 5i + 4j + 12k
Resolution of Vectors into
• Vectors describing real conditions are rarely
written in the easy-to-use Cartesian
notation. More often a vector is expressed
as an angle.
• For example, the velocity of a projectile
might be given as 30 m/s at 25o above the
• In order to work with this velocity vector in
a meaningful context we must often
“resolve” it, or break it down into its
• Generally we will employ the use of
trigonometry to accomplish this task.
V = Vx + Vy
• To find the angle formed when two
independent (i.e., x- and y- dimension)
vectors are added simply use the ratio of
their magnitudes and the tangent function.
Known Vx & Vy
• Since we are using vectors as variables to
analyze motion it is critical to note that
vectors that are at right angles to each other
• This means that motion in any one direction
does not affect motion at right angles to that
– For example: Up/Down (Vertical) motion is
independent of Side/Side (Horizontal) motion
during projectile flight.
Products of Vectors
• The Scalar Product
– The “dot” product
The scalar product of two vectors is the product of the
magnitude of one vector and the component of a second
vector along the direction of the first.
The scalar or dot product takes the form:
a b = a b cos
A B A BA A B cos
The Vector Product
• The cross product of two vectors is given by
R A B sin
where is the smallest angle between the two
• The direction of the resultant vector is
perpendicular to the plane defined by
vectors A & B and given by the right hand
In what direction is R?