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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Please read the 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.

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.