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3-Dimensional vectors

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Intros
Lessons
  1. 3 Dimensional Vectors Overview:
  2. Review of 2D vectors
    • A vector with 2 components
    • Adding and Subtracting
    • Scalar Multiplication
    • What do the vectors look like visually?
  3. 3D Vectors
    • A vector with 3 components
    • How to add and subtract
    • Scalar multiplication
    • How to visualize 3D vectors
  4. Obtaining a Vector & Calculating Length of Vector
    • Calculating a vector with 2 points: B−AB-A
    • Formula for the length of 3D vectors: a2+b2+c2\sqrt{a^2 + b^2 + c^2}
  5. Properties of Vectors
    • Adding/Subtracting
    • Scalar Multiplication
    • Distribution
    • Etc
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Examples
Lessons
  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.

    1. 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.

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

            1. 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

              Topic Notes
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              Notes:

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

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

              To add two vectors, we add the corresponding components.

              v⃗+u⃗=  <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.

              v⃗−u⃗=  <a−d,b−e,c−f>\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.

              cu⃗=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,

              B−A=  <x1−x0,y1−y0,z1−z0> B-A = \; <x_1-x_0,y_1-y_0,z_1-z_0>
              =BA⃗=\vec{BA}


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

              ∣v⃗∣ =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.