Transcript Section 6.6 - Shelton State Community College

Section 6.6
Vectors
Overview
• A vector is a quantity that has both magnitude
and direction.
• In contrast, a scalar is a quantity that has
magnitude but no direction.
Vector Representation
• A vector is usually represented by a directed
line segment, one that has an initial point and
a terminal point.
• Vectors are written using a boldface letter, or
an arrow over a single letter:

v
Magnitude
• The magnitude of a vector is its length. Use
the formula for the distance between points
to find the length of a vector:
v 
x2  x1    y2  y1 
2
2
• Two vectors are equal if they have the same
magnitude and the same direction.
Example.
• Given vector v with initial point P(5, -2) and
terminal point Q(-3, -4):
1. Sketch v.
2. Find the magnitude of v.
Unit Vectors
• A unit vector is a vector with a magnitude of
1.
• Vector i is the unit vector whose initial point is
at the origin and whose direction is along the
positive x-axis.
• Vector j is the unit vector whose initial point is
at the origin and whose direction is along the
positive y-axis.
More…
• Vectors in the rectangular coordinate system
can be represented in terms of i and j:
• If vector v has initial point at the origin and
terminal point (a,b), then
 

v  ai  bj
• a is the horizontal component and b is the
vertical component, and

v  a2  b2
More…
• If the initial point of v is not at the origin, then



v  x2  x1 i   y2  y1  j
Examples
• Let v be the vector from initial point P(-3, -5)
to terminal point Q(3, 4).
1. Sketch the graph.
2. Find the magnitude of v.
3. Write v in terms of i and j.
Vector Arithmetic in Terms of i and j






• If v  a1i  b1 j and w  a2i  b2 j
and k is a real
number then:


 
v  w  a1  a2 i  b1  b2  j


 
v  w  a1  a2 i  b1  b2  j



kv  ka1 i  kb1  j
Examples
• Let u = 2i – 7j and v = -4i + 8j. Find each of the
following vectors, written in terms of i and j.
1. u – v
2. 7u + 5v
3. The magnitude of v – u
Unit Vectors re-visited
• For any nonzero vector v, the vector

v

v
is the unit vector that has the same direction as
v.
Example
• Find the unit vector that has the same
direction as the vector v = 6i + 8j
Writing a vector in terms of its
magnitude and direction
 

 
v  v cosi  v sin j
Example: if vector v has a
magnitude ||v|| = 32 and a
direction θ = 225°, write v in
terms of i and j.
Resultant Forces
• When two vectors are acting simultaneously
on an object, the resultant force can be found
by:
1. Writing each vector in terms of i and j, then adding
the vectors together (parallelogram method).
2. Drawing the vectors from “tip to tail”, then using
the Law of Sines and/or the Law of Cosines (tip to
tail method) to find the magnitude and direction of
the resultant force.
Some Pictures
Examples
• The magnitude and direction of two forces
acting on an object are 110 pounds, S61°E,
and 120 pounds, N54°E, respectively. Find the
magnitude and direction of the resultant
force.
• MLP, Problem 15.