3-Dimensional vectors

Notes:

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

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

Where

a

represents the

x

-coordinate

b

represents the

y

-coordinate

c

represents the

z

-coordinate

Let

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

and

\vec{u} = <d,e,f>

, and

c

be a scalar.

To add two vectors, we add the corresponding components.

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

To subtract two vectors, we subtract the corresponding components.

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

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

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

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,

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

Length of a Vector
Suppose we have a vector

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

. 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:

This applies to vectors in any dimension.

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