6.3 Vectors in the Plane

6.3 Vectors in the Plane

Check out the following web sites:
http://www.math.duke.edu/education/ccp/materials/mvcalc/vectors/vec1.html

Vectors are used to analyze numerous aspects of everyday life.
Vector in the plane - given vector PQ written bold faced v = PQ (with a partial ray over the PQ but with only the top part of the arrow and with the letters PQ under it)
Vectors are denoted by lowercase, boldface letters such as u, v, and w.

Vectors are a directed line segment vector PQ having initial point P and terminal point Q.
It has magnitude, or length, is denoted by ll PQ ll.

If the directed line segment whole initial point is the origin, then the representative of the vector v is in standard position. The component form of a vector v, written v = <v₁, v₂>

otherwise:

If the directed line segment initial oint is P and the terminal point is Q then
P = (p₁, p₂) and Q = (q₁, q₂) then vector PQ = <q₁ - p₁, q₂ - p_2> = <v₁, v₂> = v

If both the initial point and the terminal point lie at the origin, v is the zero vector and denoted by 0 = <0, 0>

Example 1: Given the initial point P (1 , 0) and the terminal point Q (3 , 4) give the component form of vector PQ

P = ( p₁, p₂) and Q = (q₁, q₂)
vector PQ = < q₁ - p₁ , q₂ - p₂> = <v₁, v₂> = v = < 3 - 1 , 4 - 0 > = < 2, 4 >

Magnitude or length of v is

ll vll = √((q₁ - p₁)² + (q₂ - p₂)² ) = √(v₁², v₂²)
If llvll = 1, v is a unit vector.

llvll = 0 if and only if v is the zero vector 0.

llvll = √((3 - 1)² + (4 - 0)² ) = √(2², 4²) = √(4 + 16) = √20
= √4√5 = 2√5

Parallelogram Law - for vector addition is the sum of vector v and vector u, v + u, is often called the resultant of vector addition, is the diagonal of a parallelogram having u and v as its adjacent sides

Definition of Vector Addition and Scalar Multiplication
Let u = <u₁, u₂> and v = <v₁, v₂>
be vectors and let k be a scalar (a real number).

Then the sum of u and v is the vector
u + v = <u₁+ v_{1 ,} u₂ + v₂>

and the scalar multiple of k time u is the vector
ku = k <u₁, u₂> = <ku₁, ku₂>

Properties of Vector Addition and Scalar Multiplication:

Let u, v, and w be vectors and let “c” and “d” be scalars. Then the following properties are true.

1. u + v = v + u Addition commutative property

2. (u + v) + w = u + (v + w) Addition associative property

3. u + 0 = 0 zero property

4. u + (-u) = 0 Additive inverse

5. c (du) = (cd) u Multiplication associative property

6. (c + d ) u = cu + du Distributive property

7. c (u + v) = cu + cv Distributive property

8. 1 (u) = u, 0 (u) = 0

9. llcvll = lcl llvll

u = unit vector = (v)divided by the magnitude of v
The vector u is called a unit vector in the direction of v.

Example 2: let u = <0, -9> and v = <-6, 10>,

a). find u + v

<0, -9> + <-6, 10> = <0 + -6, -9 + 10> = <-6, 1>

b). find u - v

<0, -9> - <-6, 10> = <0 - ^-6, -9 - 10> = <6, -19>

c). find 2u - 3v

2 <0, -9> - 3<-6, 10> = <0, -18> + <18, -30> = <0 + 18, -18 - 30> = <18, -48>

d). find v + 4u

<-6, 10> + 4<0, -9> = <-6, 10> + <0 , -36> = <-6 + 0, 10 + -36> = <-6, -26>

Standard Unit Vectors
i = <1, 0> and j = <0, 1>

Given v = <v₁, v₂>
v = v₁ <1, 0> + v₂ <1, 0>
= v₁ i + v₂ j

is the Linear Combination of the vectors i and j

Writing a Linear Combination of Unit Vectors
Example 3: Given initial point P (5, -12) and terminal point Q (-4, 7)

Begin by writing the component form of the vector PQ

<5 + -4, -12 + 7> = <1, -5>
Then to write vector PQ as a linear combination of the standard unit vectors i and j
vector PQ = 1i -5j

Example 4: let u = 2i - j and v = -i + j
u + v = 2i - j + -i + j = i so therefore u + v = <1, 0>

to check, recall i = <1, 0> and j = <0, 1> so therefore

u = 2 <1, 0> - <0, 1> = <2, 0> - <0, 1> = <2, -1>
v = - <1, 0> + <0, 1> = <-1, 0> + <0, 1> = <-1, 1>
so u + v = <2, -1> + <-1, 1> = <2 + -1, -1+ 1> = <1, 0>
same answer

Example 5: Vector Operations:
let u = -2i + 3j and v = 4i + 5j
Find 2u - 3v

2(-2i + 3j) - 3(4i + 5j) = -4i + 6j - 12i -15j
= -4i -12i + 6j -15j = -16i -9j