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

$\vec{v} = $

Where
$a$ represents the $x$-coordinate
$b$ represents the $y$-coordinate
$c$ represents the $z$-coordinate

Let $\vec{v} = $ and $\vec{u} = $, and $c$ be a scalar.

$\vec{v} + \vec{u} =\; $

To subtract two vectors, we subtract the corresponding components.

$\vec{v} - \vec{u} =\; $

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

$c\vec{u} = c\; = \;$

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

$B-A = \; $
$=\vec{BA}$

Length of a Vector
Suppose we have a vector $\vec{v} =\;$. Then the length of the vector will be the formula:

$|\vec{v}|\ = \sqrt{a^2+b^2+c^2}$

Vector Properties Let $v,u,w$ be vectors and $c$ be a scalar. Here are the following vector properties:
1. $u+v=v+u$
2. $(u+v)+w=u+(v+w)$
3. $c(u+v) = cu+cv$
4. $(c+d)u=cu+du$
5. $c(du) = (cd)(u)$
6. $1u=u$
This applies to vectors in any dimension.
• Introduction
3 Dimensional Vectors Overview:
a)
Review of 2D vectors
• A vector with 2 components
• 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
• Scalar Multiplication
• Distribution
• Etc

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

Calculate $2u+3v$.

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

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

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

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

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

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

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

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