Vectors: 5 Minute Review

Vectors can be added or subtracted.
◦ To add vectors graphically, draw one after the
other, tip to tail.
◦ To add vectors algebraically,
 Resolve the vectors into components.
 Add the components of each direction.
 Use the Pythagorean Theorem to find the
magnitude of the resultant vector and inverse tan
to find the angle.
Vector Addition Example

A boy pushes horizontally on a wagon
with a force of 10 N. A girl pulls on the
wagon’s handle at an angle of 30 degrees
from the horizontal with a force of 8 N.
What is the net force acting on the
wagon?
Unit Vectors

A unit vector is a vector that points along
the x, y or z axis and is one unit long.
◦ The symbols for the x, y, and z unit vectors
are
ˆi, ˆj, and kˆ
◦ Any vector can be expressed as a sum of its x,
y and z components multiplied by unit
vectors.
◦ The dimensions

 of the quantity are stated
thej unit vectors, e.g.
v along
3 m/s iwith
 2 m/s
©2008 by W.H. Freeman and Company
Unit Vector Example

A car has a velocity of 6 m/s in a direction
30o north of east. Express this vector in
terms of unit vectors. Let east be the
positive x direction and north the positive
y direction.
Operations with Vectors: Magnitude

To find the magnitude of a vector, use the
Pythagorean Theorem.
  
If r  x i  y j  zk
then r  r  x  y  z
2
2
2
Operations with Vectors: Addition
and subtraction

To add two vectors, add their
components.
 
v1  3 i  2 j
 
and v 2  4 j  5k
  
v r  v1  v 2  3 i  6 j  5k
Vector Addition & Subtracting
Graphical method

Addition: Connect head to tail

Subtraction: flip the subtrahend 180°, then
connect head to tail.
Multiplication of Vectors:
1. Vector Product of a Scalar and a
Vector
2. Scalar Product of Two Vectors
3. Vector Product of Two Vectors
Multiplication of Vector by Scalar
• Applications
• momentum p = mv
• electric force F = qE
• Result
• A vector with the same direction, a
different magnitude and perhaps
different units.
Multiplication of Vector by Vector
(Dot Product)
• Application
• work W = F  d
• Result
• A scalar with magnitude and no
direction.
Multiplication of Vector by Vector
(Dot Product)
C=AB
C = AB cos 
A

B
Dot Product Practice
i • i = 1x1x cos0
i • j = 1x1x cos 90
i • k = 1x1x cos 90
j • j = 1x1x cos0
j • k = 1x1x cos 90
j • i = 1x1x cos 90
k • i = 1x1x cos 90
k • k = 1x1x cos0
k • j = 1x1x cos 90
5i • 3i =
4i • 2j =
-4i • 2k =
5j • -3j =
2j •2 k =
1j •9 i =
9k •7 i =
-√3k • -2k =
½k • √2j =
Multiplication of Vector by Vector
(Cross Product)
• Application
• Work
=rF
• Magnetic force F = qv  B
• Result
• A vector with magnitude and a
direction perpendicular to the plane
established by the other two
vectors.
Multiplication of Vector by Vector
(Cross Product)
C=AB
C = AB sin  (magnitude)
Direction determined by
Right Hand Rule
A

B
Multiplication of Vector by Vector
(Cross Product)
C=AB
A

B
Cross Product Practice
i x i = 1x1x sin0
i x j = 1x1x sin90
i x k = 1x1x sin90
j x j = 1x1x sin0
j x k = 1x1x sin90
j x i = 1x1x sin90
k x i = 1x1x sin90
k x k = 1x1x sin0
k x j = 1x1x sin90
5i x4i =
3i x 3j =
-9i x 6k =
5j x5 j =
√3j x √3k =
2.4j x 2 i =
½k x ¾i =
-2k x 2k =
-2k x 2j =
Vector Multiplication:
2 types: Cross product & dot product