1. Home
  2. >Calculus 3
  3. >Three Dimensions
Help

3-Dimensional vectors

Everything You Need in One Place

Homework problems? Exam preparation? Trying to grasp a concept or just brushing up the basics? Our extensive help & practice library have got you covered.

Learn and Practice With Ease

Our proven video lessons ease you through problems quickly, and you get tonnes of friendly practice on questions that trip students up on tests and finals.

Instant and Unlimited Help

Our personalized learning platform enables you to instantly find the exact walkthrough to your specific type of question. Activate unlimited help now!

0/4
?
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
0/6
?
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
              ?
              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.