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

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Answer 1 · 2018-08-04T01:15:33

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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.

enter image source here

$vec(OA) = hat(u)=(2 hat i+5 hat j)$

In component form

$hat(u)=<2,5>$

$vec(OB) = hat(v)=(4 hat i-8 hat j)$

In component form

$hat(v)=<4,-8>$

Let us see how we can add these two vectors:

$hat (u) + hat (v) = (2 hat i+5 hat j)+(4 hat i-8 hat j)$

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:

enter image source here

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:

$vec(OA)$ can also be translated to the line in green (BC).

OR

$vec(OB)$ can be translated to the line in blue (AC).

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

Hope it helps.