# Component Vectors

## Key Questions

$\text{ }$

#### Explanation:

$\text{ }$
How do we use the components of two vectors to find the resultant vector by adding the two vectors ?

A Vector is defined as a quantity with both magnitude and direction.

Two vectors are shown below:

color(red)(vec(OA) and vec(OB)

We will also be using these vectors in our example later.

$\vec{O A} = \hat{u} = \left(2 \hat{i} + 5 \hat{j}\right)$

In component form

$\hat{u} = < 2 , 5 >$

$\vec{O B} = \hat{v} = \left(4 \hat{i} - 8 \hat{j}\right)$

In component form

$\hat{v} = < 4 , - 8 >$

Let us see how we can add these two vectors:

$\hat{u} + \hat{v} = \left(2 \hat{i} + 5 \hat{j}\right) + \left(4 \hat{i} - 8 \hat{j}\right)$

Using component form:

$\hat{u} + \hat{v} = < 2 , 5 > + < 4 - 8 >$

Add color(red)(i components and color(red)(j components together:

$\hat{u} + \hat{v} = < 2 + 4 > + < 5 - 8 >$

color(red)(hat (u) + hat (v) =<6, -3>

We can represent this solution graphically as follows:

The solution is represented by

color(red)(w=hat (u) + hat (v) =<6, -3>

OR

color(red)(w=hat (u) + hat (v) =(6i -3j)

Note: Alternative graphical solution process:

$\vec{O A}$ can also be translated to the line in green (BC).

OR

$\vec{O B}$ can be translated to the line in blue (AC).

We can see that color(red)(w is the solution.

Hope it helps.

• To find the magnitude of a vector using its components you use Pitagora´s Theorem.

Consider in 2 dimensions a vector $\vec{v}$ given as:
$\vec{v} = 5 \vec{i} + 3 \vec{j}$ (where $\vec{i}$ and $\vec{j}$ are the unit vectors on the x and y axes)

The magnitude of this vector (or its length in geometrical sense) is given using Pitagora's Theorem, as:
$| \vec{v} | = \sqrt{{5}^{2} + {3}^{2}} = 5 , 8$

The same thing applies in 3 dimensions, the only thing is to include the third component.

So if the vector is now given as:
$\vec{v} = 5 \vec{i} + 3 \vec{j} + 2 \vec{k}$
The magnitude will be:
$| \vec{v} | = \sqrt{{5}^{2} + {3}^{2} + {2}^{2}} = 6 , 2$

• Often when two processes interact we only know the component vector values and need to be able to combine these to get a desired result.

This might be more easily understood by an example:
Suppose I am trying to fly from point A to point B which is due North of point A. My plane flies at an air speed of 100 miles/hour but there is a wind blowing due West at 30 miles/hour. How many degrees East of North do I need to orient my plane to fly in a straight line to B?

From the above diagram, I need to head my plane (approximately)
${17.5}^{o}$ East of North.

This problem could be extended to ask:
If it is 200 miles from A to B and my plane has enough gas to fly 250 miles will I be able to make this trip?

• A vector has both magnitude (which is its length) and direction (which is its angle).

Any two dimentional vector at an angle will have a horizontal and a vertical component .
A vector written as ( 12 , 8 ) will have 12 as its horizontal component, and 8 as its vertical component, and because both components are positive, the vector is pointing to the northeastern direction.