vectory

R3 equation of a sphere with center C(h,k, l) and radius r =

(x-h)^2+(y-k)^2+(z-l)^2=r^2 if the center of sphere is at the origin O, then an equation of the sphere is x^2+y^2+z^2=r^2 if the sphere is a ball, use ≤ r^2

cos(θ) > 0 if and only if

0 ≤ θ ≤ π/2

Properties of Dot Product (there are five) a, b, and c are vectors. d is a scalar.

1. a•a= IaI^2

1. a+b=b+a

2. a+(b+c) = (a+b)+c associative law

2. a•b=b•a

3. a•(b+c)=ab+ac

3. a+0=a

4. a+(-a)=0

4. (da)b=(db)a=(ab)d

5. a•0=0

5. c(a+b)=ca+cb

6. (c+d)a= ca+da

7. (cd)a=c(da)

8. 1•a=a

A position (or displacement) from a reference point can be indicated by an arrow with its tail at the reference point and its head at the position.

A displacement is the shortest distance from the initial to the final position of a point P.[1] Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P. A displacement vector represents the length and direction of that imaginary straight path.

The magnitude or length of the vector "v" is the length of any of its representations and is denoted by the symbol IvI or IIvII.

By using the distance formula to compute the length of segment OP, we obtain the following formulas. The length of the R2 vector "a"=< a1, a2 > is |a|=√((a1)^2+(a2)^2) The length of the R3 vector "a"=< a1, a2, a3 > is |a|=√((a1)^2+(a2)^2+(a3)^2)

In an extreme case where a and b point in exactly the same direction, we have x=0, so cos(θ)=1 and a*b=IaI*IbI

If a and b point in exactly opposite directions, then θ = π and so cos(θ) = -1 and a*b= (-)IaI*IbI

Properties of vectors: The following eight properties can be verified algebraically or geometrically.

If a, b, and c are vectors in Vn and c and d are scalars, then:

Multiplication of a vector by a real number: In this context we call the real number "c" a scalar to distinguish it from a vector. 2< a1, a2 > = < a1, a2 >+<a1, a2>=< 2a1, 2a2 >

In general, we multiply a vector by a scalar by multiplying each component by that scalar.

Three vectors in V3 play a special role.

Let: i=< 1,0,0 >, j=< 0,1,0 >, k=< 0,0,1 > then i, j, and k are vectors that have length 1 and point in the directions of the positive x, y, and z axes. Similarly, in two dimensions we define i=< 1,0 > and j=< 0,1 >.

The term vector is used to indicate what?

That a quantity (such as velocity or force) has both a magnitude and direction.

The "direction angles" of a nonzero vector a are the angles:α, β, γ in the interval [0, π] that a makes with the positive x-, y-, and z-axes.

The cosines of these direction angles cos(α), cos(β), and cos(γ), are called the "direction cosines" of the vector a.

We can think of a*b as measuring the extent to which a and b point in the same direction.

The dot product a*b is positive if a and b point in the same general direction, 0 if they are perpendicular, and negative if they point in generally opposite directions.

A vector is often represented by an arrow or a directed line segment.

The length of the arrow represents the magnitude of the vector and the arrow points in the direction of the vector.

Dot Product

The product of two multiplied vectors. Dot product is not a vector. It is a real number or a scalar. Dot product may be called a scalar product or an inner product.

The only vector with length zero (0) is the zero vector 0=<0,0> (or 0=< 0,0,0 >).

This vector is also the only vector with no specific direction.

Based on the perpendicular angle relation to vectors we have the following theorem.

a*b=IaI*IbI*cos(π/2)=0. and conversely... if a*b=0, then cos(θ)=0, so theta = π/2.

By the difference a-b of two vectors, we mean

a-b=a+(-b) so if a=< a1, a2 > and b=< b1, b2 >, then a-b=< a1-b1, a2-b2 >

Find the vector represented by the directed line segment with initial point A(2, -3, 4) and terminal point B(-2, 1, 1).

a=< -2-2, 1--3, 1-4 > =< -4, 4, -3 >

if a=< a1, a2, a3 >, then we can write:

a=< a1, a2, a3 > =< a1, 0, 0 >+< 0, a2, 0 >+< 0, 0, a3 > =a1< 1,0,0 >+a2< 0,1,0 >+a3< 0,0,1 > ergo: a=a1i+a2j+a3k Thus any vector in V3 can be expressed in terms of the "standard basis vectors i, j, and k".

a and b are orthogonal if and only if...

a•b=0.

Example. If the vectors a and b have lengths 4 and 6 and the angle between them is pi/3, find a•b.

a•b=IaI•IbI•cos(pi/3) =4•6•(1/2) =12

Theorem if theta is the angle between the vectors a and b, then...

a•b=IaI•IbI•cos(θ)

Two nonzero vectors "a" and "b" are called parallel if...

b=c(a) for some scalar c.

If c is a scalar and "a"=< a1, a2 > then the vector c"a" is defined by...

c"a"=< ca1, ca2 >

Corollary If theta is the angle between the nonzero vectors a and b, then...

cos(theta)= a•b/ (IaI•IbI) remember IaI=√((a1)^2+(a2)^2+(a3)^2)

Using the previous Corollary, but replacing the b with i, we have the following...

cosα = (a*i)/IaI*IiI = a1/IaI similarly, we also have: cosβ = a2/IaI and cosγ = a3/IaI. squaring these expressions we have: cos^2α + cos^2β + cos^2γ = 1. This can be written: IaI< cos α, cos β, cos γ >

Dot Product

given a=< 1, 2, 3 > and b= < 3, 3, 3 > the dot product = 1*3+2*3+3*3= 18

Consider the scalar multiple c"a". If "a"=< a1, a2 > then, |c"a"|=√((ca1)^2+(ca2)^2) =√(c^2(a1)^2+(a2)^2) =√c^2•√((a1^2)+(a2^2))= |c|•|a| ...so the length of c"a" is |c| times the length of "a".

if a1≠0, we can talk about the slope "a" as being a2/a1. But, if c≠0, then the slope of c"a" is ca2/ca1 = a2/a1, the same as the slope of "a". the vector -a=(-1)•a. it has the same length as "a" but it points in the opposite direction.

because of the nature of the value of cosine being governed by its relation to π/2

insofar as it has bearing on the dot product we can see that a*b is positive for all θ between 0 and π/2 and negative for all angles between π/2 and π.

Two nonzero vectors a and b are called... if the θ between them is theta = π/2.

perpendicular or orthogonal

The zero vector is considered to be...

perpendicular to all vectors. Therefore we have this method for determining whether two vectors are orthogonal.

if a = IaI< cos α, cos β, cos γ > then, a*(1/IaI)=< cos α, cos β, cos γ >

this says that the direction cosines are the components of the unit vector in the direction of a.

Observe that if α=< a1, a2, a3 > is a vector where the initial point is A(x1, y1, z1) and the terminal point is B(x2, y2, z2) then...

we must have: x1+a1 = x2, y1+a2 = y2, z1+a3=z2 and so a1=x2-x1, a2=y2-y1, a3=z2-z1... Thus given the points A(x1,y1,z1) and B(x2, y2, z2) the vector with representation AB→ is α=<x2-x1, y2-y1, z2-z1>

The Cartesian product R x R x R =

{(x,y,z)|x,y,z ∈ R} is the set of all ordered triples of real numbers and is denoted by R^3. We have given a one-to-one correspondence between points P in space and ordered triples (a,b,c) in R^3. It is called a three-dimensional rectangular coordinate system. Notice that, in terms of coordinates, the first octant can be described as the set of points whose coordinates are all positive.

cos(θ) < 0 if and only if

π/2 ≤ θ ≤ π

A 2-dimensional vector is

an ordered pair α=< a1,a2 > of real numbers

R3 distance formula: Between points p1 (x1,y1,z1) and p2(x2,y2,z2)

|p1p2|=√[(x2-x1)^2+(y2-y1)^2+(z2-z1)^2]

We add vectors by adding the corresponding components of two vectors.

If "a"=< a1,a2 > and "b"=< b1, b2 >, then the vector "a"+"b" is defined by: "a"+"b"=< a1+b1, a2+b2 > this goes on ... < a1+b1, a2+b2, a3+b3, a4+b4,...a(n-1)+b(n-1), a(n)+b(n)+a(n+1)+b(n+1)... >

We denote by V2 the set of...

all two-dimensional vectors. And by V3 we denote the set of all three-dimensional vectors. More generally, we will later need to consider the set Vn of all n-dimensional vectors. An n-dimensional vector is an ordered n-tuple: a=< a1, a2, ..., an >

A 3-dimensional vector is

an ordered pair α=< a1,a2, a3 > of real numbers. The numbers a1,a2, a3 are called the components of α.