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 = ku1,u2=ku1,ku2

Geometric representation of vector addition

u

v

u+v

Tail-to-head

u

v

u+v

parallelogram

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 2u ½u-u-½u

ExampleLet 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 = -1,7Geometrically: start with -3,2 and move right 2 and up 5

Exampleb) Using component form definition of scalar:

2u = 2-3,2=-6,4

Examplec) 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 + v

u + (-1)v

u – w

3v

1,5 2,3 4, 7 u v w

1,5 2,3 1,8

1,5 ( 1) 2,3 1,5 2, 3 3,2

1,5 4, 7 3,12

3 2,3 6,9

Example

2u + 3w

2u – 4v

-2u – 3v

-u – v

1,5 2,3 4, 7 u v w

2 1,5 3 4, 7 2,10 12, 21 14, 11

2 1,5 4 2,3 2,10 8,12 10, 2

2 1,5 3 2,3 2, 10 6,9 4, 19

1,5 2,3 1, 5 2,3 1, 8

Homework

Pg 511: 9-20 all