Section 1.1
Vectors
The vector from point A ( x 1 , y 1 ) to the point B ( x 2 , y 2 ) is

AB  x 2  x 1 , y 2  y 1  run , rise .
Note that infinitely many pairs of points result in the same vector.
Scalar: In a discussion of vectors, ordinary numbers are called scalars.
Components: The components of the vector a , b are the scalars a
and b .
A vector can be thought of as a set of directions for moving from point
A to point B.



Example: Find the vectors AB , BA and CD for the points
A (  1, 2 ), B ( 3 , 5 ), C ( 2 , 7 ) and D ( 6 ,10 ) .
Length of a vector: The length of the vector a , b is
a,b

a b .
2
2

Note that AB is the distance from point A to point B.
Scalar Multiplication: c a , b  ca , cb .


Note that BA   AB .
A unit vector is a vector of length 1.
Example: Find a unit vector with the same direction as 3 ,  4 .
Then find two vectors of length 2 that are parallel to 3 ,  4 .
Adding Vectors: a 1 , b1  a 2 , b 2  a 1  a 2 , b1  b 2 .




Examples: 1) Sketch v  1, 3 and w  4 ,  1 . Sketch v  w
2) Sketch a parallelogram with adjacent edges formed by the vectors


v  1 , 3 and w  4 ,  1 . What are the diagonals?
3) The parallelogram PQRS has adjacent edges PQ and PS
for
P (1, 2 ), Q ( 2 , 5 ), and S ( 5 ,1) . Find the vertex R.

Basic Unit Vectors: The vectors i  1, 0 and

j  0 ,1 are the basic


unit vectors. Any vector a , b can be written as a i  b j .

Base Angle of a Vector: For a , b  OA for O ( 0 , 0 ), A ( a , b ) Let

be the angle between a , b  OA and the positive x-axis. Then

a,b 



a  b  cos  i  sin  j  .


2
2
Example: A 50 lb weight hangs from a ceiling by two strings. String 1
makes a 30 degree angle with the ceiling to the left and String makes a
45 degree angle to the right. The weight is at rest. Find the tensions in
the strings.