3-Dimensional vectors

3-Dimensional vectors

Lessons

Notes:

3 Dimensional Vectors
3 Dimensional vectors have three components instead of two. 3D vectors are in the form

v=<a,b,c>\vec{v} = <a,b,c>

Where
aa represents the xx-coordinate
bb represents the yy-coordinate
cc represents the zz-coordinate

Let v=<a,b,c>\vec{v} = <a,b,c> and u=<d,e,f> \vec{u} = <d,e,f>, and cc be a scalar.

To add two vectors, we add the corresponding components.

v+u=<a+d,b+e,c+f>\vec{v} + \vec{u} =\; <a+d,b+e,c+f>

To subtract two vectors, we subtract the corresponding components.

vu=<ad,be,cf>\vec{v} - \vec{u} =\; <a-d,b-e,c-f>

To multiply a scalar with a vector, we multiple the scalar to each individual component.

cu=c<d,e,f>=<cd,ce,cf>c\vec{u} = c<d,e,f>\; = \;<cd,ce,cf>


Obtaining a Vector with 2 points.
Let the two points be A=(x0,y0,z0)A=(x_0,y_0,z_0) and B=(x1,y1,z1)B=(x_1,y_1,z_1). Then we can create a vector between these points by subtracting them. In other words,

BA=<x1x0,y1y0,z1z0> B-A = \; <x_1-x_0,y_1-y_0,z_1-z_0>
=BA=\vec{BA}


Length of a Vector
Suppose we have a vector v=<a,b,c>\vec{v} =\;<a,b,c>. Then the length of the vector will be the formula:

v =a2+b2+c2 |\vec{v}|\ = \sqrt{a^2+b^2+c^2}


Vector Properties Let v,u,wv,u,w be vectors and cc be a scalar. Here are the following vector properties:
  1. u+v=v+uu+v=v+u
  2. (u+v)+w=u+(v+w)(u+v)+w=u+(v+w)
  3. c(u+v)=cu+cvc(u+v) = cu+cv
  4. (c+d)u=cu+du(c+d)u=cu+du
  5. c(du)=(cd)(u)c(du) = (cd)(u)
  6. 1u=u1u=u
This applies to vectors in any dimension.
  • Introduction
    3 Dimensional Vectors Overview:
    a)
    Review of 2D vectors
    • A vector with 2 components
    • Adding and Subtracting
    • Scalar Multiplication
    • What do the vectors look like visually?

    b)
    3D Vectors
    • A vector with 3 components
    • How to add and subtract
    • Scalar multiplication
    • How to visualize 3D vectors

    c)
    Properties of Vectors
    • Adding/Subtracting
    • Scalar Multiplication
    • Distribution
    • Etc


  • 1.
    Adding, Subtracting, and Scalar Multiplication of Vectors
    Let two vectors be u=<2,0,5> u = <2,0,5> and v=<3,2,1> v = <3,2,-1> .

    Calculate 2u+3v2u+3v.


  • 2.
    Let two vectors be u=<1,2,4>u = <-1,2,-4> and v=<1,5,3> v = <1,-5,-3> .

    Calculate 3(u+v)v3(u+v) - v.


  • 3.
    Vectors From 2 Points
    Create a vector from the two points: A=(1,4,5)A = (-1,4,5) and B=(4,2,4) B = (4,2,-4) .

  • 4.
    Length of a Vector
    Let P=(2,5,3)P=(2,5,3) and Q=(2,1,6)Q=(-2,1,6). Find PQ\vec{PQ} and its length.

  • 5.
    Verifying Properties of Vectors
    Use the two vectors u=<3,1,5>u=<3,1,5> and v=<1,4,6>v=<1,4,-6>, and the scalar to show that:

    c(u+v)=cu+cvc(u+v) = cu + cv


  • 6.
    Use the two scalars c=1c=1 and d=3d=3 and the vector v=<1,4,6>v=<1,4,-6> to show that:

    (c+d)v=cv+dv(c+d)v = cv + dv