Vectors
A vector is a directed line segment, which has both a
magnitude (length) and direction.
A vector can be created using any two points in the
plane, the direction of the vector is usually denoted
by the placement of an arrow at the end of the line
segment.
ex) Pictured here is a vector (named v ) which has its initial point (‘tail’ point) at
(−2, −1) and its terminal point (the arrow ‘head’) at (2, 3) .
The component form of a vector is written using the form v = ai + bj .
‐ The value of a is always written as a coefficient of i ... which represents the
vector’s horizontal component.
‐ The value of b is always written as a coefficient of j ... which represents the
vector’s vertical component.
ex) What is the component form of the vector shown in the picture above?
The magnitude of a vector (which is denoted as v ) is simply its length.
It is calculated by applying the Pythagorean Theorem to the component
coefficients a and b .
For any vector v = ai + bj , its magnitude is v = a2 + b2
ex) What is the magnitude of the vector from the picture above? (Use the
component form)
ex) Find the component form and the magnitude of the vector v with initial point
at (−3, 11) and terminal point at (9, 40) . (Approximate the magnitude to 2 decimal places.)
The two main operations with vectors are vector addition and scalar
multiplication. These operations can be done algebraically and graphically.
Ex) For the vectors u = −i + 3j and v = −2i − j
Plot vectors with their initial points at the origin and determine the following
vector combinations.
(a) u + v
Find the sum algebraically
(using component forms)
Calculate u + v
Find the sum graphically
(using parallelogram law)
(b) 2u + 3v
Find the sum algebraically
(using component forms)
Find the sum graphically
(using parallelogram law)
The ‘2’ applied to u and the
number ‘3’ applied to v are
examples of scalar multiplication.
The scalars will ‘scale’ the length
of each vector making them longer.
Calculate 2u + 3v
(c) u − v
Find the sum algebraically
(using component forms)
Calculate u − v
Find the sum graphically
(using parallelogram law)
Direction Angle for a Vector
The direction angle is always considered to be the standard position angle starting
on the positive x‐axis rotating counterclockwise to the vector’s position. It can be
determined by the formula:
tanθ =
horizontal component
vertical component
To get the angle you’ll need to use TAN−1 on your calculator ... BUT MAKE SURE
YOUR ANGLE IS LOCATED IN THE CORRECT QUADRANT!
ex) Determine the direction angle for the vector v = 5i + j .
(IT ALWAYS HELPS TO SKETCH THE VECTOR FIRST)
ex) Determine the direction angle for the vector v = −4i + 7 j .
(SKETCH THE VECTOR FIRST AND BE CAREFUL USING TAN−1 )
Unit Vectors
When you want to preserve the direction of a certain vector but apply a different
length to it, you’ll need to transform that vector in to a unit vector ... essentially a
vector of length 1.
To get a unit vector, you divide the components by the magnitude:
v
unit vector =
v
ex) What is the unit vector which has the same direction as v = 4i − 3j ?
ex) A force of 1200 lbs is applied in the direction of the vector v = 4i − 3j .
What are the components of this force vector? (Call the force vector F)
Decomposing a Vector
When you already know the magnitude v and direction angle, θ , of a vector v ,
you can write it in component form using the formulas
Horizontal Component Æ v ⋅ cosθ
Vertical Component Æ
v ⋅ sinθ
Creating the vector v = ( v ⋅ cosθ )i + ( v ⋅ sinθ )j
ex) Find the horizontal and vertical components of the vector with a length of
v = 800 and a direction ofθ = 145°. (Round components to 2 decimal places).
ex) A jet is flying in a direction of N 20°E with a speed of 500 mi/h. Represent the
velocity of this jet as a vector in component form. (2 decimal place rounding)
Resultant Vectors
The resultant of two or more vectors is the ‘result’ of all of the vectors acting on
the same object at the same time.
A resultant is simply their vector sum.
ex) Two tugboats are pulling a barge due north through a channel. One tugboat is
pulling with a force of 3500 lbs at a heading of N 20° E and the other tugboat is
pulling with 4000 lbs of force at a heading of N 25° W. (See the diagram.)
a) What are the component forms of the force vector for each tugboat?
b) Calculate the resultant vector.
This is the vector sum of adding
the tugboat vectors together.
c) What is the magnitude and the
direction of the resultant vector?
Give the direction as a bearing.