Today in Precalculus
• Notes: Vector Operations
• Go over homework
• Homework
Vector Operations
• Vector Addition
• Vector multiplication (multiplying a vector by a scalar
or real number)
Let u= u1,u2 and v= v1,v2 and k be a real
number (scalar). Then:
1. The sum of vectors u and v is the vector
u+v = u1,u2+
v1,v2=u1+v1,u2+v2
2. The product of the scalar k and the vector
u =ku = ku1,u2=ku1,ku2
Geometric representation of
vector addition
v
u
u+v
u+v
u
parallelogram
Tail-to-head
v
Geometric representation
of vector multiplication
The product ku can be represented by a stretch or
shrink of u by a factor of k when k>0. If k<0, then u
also changes direction.
u
-½u
-u
2u
½u
Example
Let u= -3,2 and v = 2,5. Find the component
form of the following vectors: a) u + v, b) 2u, c) 3u-v
a) Using component form definition of sum of vectors:
u + v =-3,2 + 2,5 = -3+2,2+5 = -1,7
Geometrically: start with -3,2
and move right 2 and up 5
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













 

 













 










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
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Example
b) Using component form definition of scalar:
2u = 2-3,2=-6,4

























Example
c) Using the component form definitions:
3u – v
= 3-3,2 – 2,5
= -9,6 – 2,5
= -11,1
or 3u + (–v)
= 3-3,2 + (–1)2,5
= -9,6 + -2,-5
= -11,1
Example
u  1,5
u+v
v  2,3
w  4, 7
 1,5  2,3  1,8
u + (-1)v
 1,5  (1) 2,3  1,5  2, 3  3, 2
u–w
 1,5  4, 7  3,12
3v
 3 2,3  6,9
Example
u  1,5
v  2,3
w  4, 7
2u + 3w
 2 1,5  3 4, 7  2,10  12, 21  14, 11
2u – 4v
 2 1,5  4 2,3  2,10  8,12  10, 2
-2u – 3v
 2 1,5  3 2,3  2, 10  6,9  4, 19
-u – v
  1,5  2,3  1, 5  2,3  1, 8
Homework
Pg 511: 9-20 all