Math 1210-007
Vectors
Fall 2010
Vectors
Definition:
A vector is a quantity which has both a magnitude and direction.
There are two equivalent ways to represent a vector quantity
1.
2.
Magnitude & Direction
Horizontal Component & Vertical Component
Think of a vector as a line segment with an arrowhead drawn at the front end. For our purposes,
we will always assume that the vector is drawn on a set of axes such that tail of the vector is
located at the origin and the head of the vector (the side with the arrowhead) is located at some
other point on the axis (see example below)
(x,y)
Let's call the vector to the left . Then we introduce the
following notation:
= Magnitude (length of the vector)
= Direction (angle with positive x-axis)
= horizontal component
= vertical component
In the figure note that the arrow head starts at the origin and ends at a point in the plane which
we denote as ( x , y ). The horizontal component of the vector is the x-coordinate of the end
point while the vertical component of the vector is the y-coordinate of the end point. Given the
magnitude and direction of a vector, one can find the horizontal and vertical components by the
following formulas:
In this course we will use vectors to describe quantities such as velocity, force and acceleration.
Given the magnitude and direction of one of these quantities
Math 1210-007
Vectors
Fall 2010
We have spent time understanding the motion of objects in one dimension. The motion of
objects in two dimensions can be understood by breaking down the motion into its horizontal and
vertical components and considering those components separately, each of which can be treated
the same as one dimensional motion.
Examples:
1.
An arrow is shot at a 55° angle above the ground with an initial velocity of 80m/s. What
are its initial vertical and horizontal components of velocity?
m/s
m/s
2.
An airplane, during lift off, is accelerating at 5 m/s2 at an angle of 20°. What are the
vertical and horizontal components of its acceleration?
m/s2
m/s2
3.
When lifting a chair up a staircase you exert on the chair a 100N force at a 45° angle.
What are the vertical and horizontal components of this force?
N
N
(Does it make sense that the horizontal and vertical forces should be equal in this case?)