# How do you find the resultant as a sum of two components?

Aug 4, 2018

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#### Explanation:

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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.