Vector Functions

Vector

quantity that has both a magnitude and a direction

Two vectors are equal if

their magnitude and direction are equal

True/False. Vector encodes starting point.

False- the change

How to find a vector?

subtract one point from another

PQ = <1,2>, what is QP?

<-1,-2> bc opposite direction so just -PQ

Scalar * Vector changes

length

-Scalar * Vector changes

direction & length

Magnitude

length of a vector

How do you denote magnitude?

l v l

How do you find magnitude?

sqrt((v₁)² + (v₂)² +… (v_n)²) with v = <v₁, v₂, … v_n>

Direction

way vector is pointing; any positive scalar multiple does not change it

Unit Vector

a vector with a length of one

True/False: There is a unit vector in every direction.

True

How do you calculate a unit vector?

take any vector and divide by its magnitude

Unit Vector in x-direction

i = <1,0,0>

Unit vector in y-direction

j = <0,1,0>

Unit Vector in z-direction

k = <0,0,1>

Cross-Product

a vector operation that inputs two vectors in 3D and outputs one vector in 3D that is orthogonal to both

Orthogonal

3D version of perpendicular

i is

x

j is

y

k is

z

True/False: Cross-products are anti-communitative.

True

(a x b) =

-(a x b); kinda like subtraction; Anticommunicative Property

(cu) x v =

c(u x v)

(u + v) x w =

(u x w) + (v x w); Distributive Property

w x (u + v) =

(w x u) + (w x v); Distributive Property

Cross Product Formula

u x v = (u₂v₃ - v₂u₃)i - (u₁v₃ - v₁u₃)j + (u₁v₂ - v₁u₂)k

Cross Product with Matrix Set Up

[ i j k ] i j

u [u_x u_y u_z ] u_x u_y

v [v_x v_y v_z] v_x v_y

True/False: Cross-Products are associative.

False bc not always

0 (zero vector)

records nonsense; too many things apply

u x v = 0

u and v are parallel (have same direction)

Direction: u x v is _____ ot u and v

orthogonal

Length: l u x v l =

l u l l v l lsin(theta)l with theta = angle between u & v

When u & v are orthogonal, then sin(theta) is

1 so lu x vl have same length

When u = v, theta = ____ and lu x vl =

zero; 0 & nonsense

Area of paralellogram with u and v

l u x v l; aka length of cross-product

Slope of a line as a vector

< 1, m >

Intercept of line as point

(0, b)

Line in space

the set of terminal points of a vector (scalar multiple) that starts at a point

Vector form of a line in direction v

r(t) = r_o + tv; requires any point and a direction

Parallel Lines in Space

have same direction

Parameterization

describe the line r(t) as components (aka x(t) = x_o + ta)

parameter

variable t

parametric equations when P = (x_o, y_o, z_o) in direction v = <a,b,c>

x(t) = x_o + ta; y(t) = y_o + tb; z(t) = z_o + tc

True/False. There is one parameterization through a line through P in the direction v.

False bc scalar could change v and many starting points which change vector forms

Check for Intersections

set equations (parametric or others) equal to each other and solve

Plane in 3D

set of all the terminal points of vectors starting at P_o that are orthogonal to n; no squares (linear terms) and might be multiplied by scalar

scalar equation of a plane w/ n = <a,b,c> & point P_o = (x_o, y_o, z_o)

0 = a(x-x_o) + b(y-y_o) + c(z-z_o)

How to solve for scalar equation

start with a point and a vector and find the dot product

Parametric equation of a place through P_o = (x_o, y_o, z_o) & vectors v₁ = <a, b, c> & v₂ = <d,e,f>

f(t,u) = <x_o, y_o, z_o> + t<a,b,c> + u<d,e,f>

x(t) = x₀ +ta + ud

y(t) = y_o + tb + ue

z(t) = z_o + tc + uf

Parallel with planes

vectors have the same normal vectors

Normal Vector

tells the direction of plane bc it is orthogonal to the plane