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

Question

11699 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-08-04T01:15:33

$" "$
Please read the explanation.

$" "$
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:

$- - \to O A and - - \to O B$ $color(red)(vec(OA) and vec(OB)$

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

enter image source here

$- - \to O A = ˆ u = (2 ˆ i + 5 ˆ j)$ $vec(OA) = hat(u)=(2 hat i+5 hat j)$

In component form

$ˆ u = < 2, 5 >$ $hat(u)=<2,5>$

$- - \to O B = ˆ v = (4 ˆ i - 8 ˆ j)$ $vec(OB) = hat(v)=(4 hat i-8 hat j)$

In component form

$ˆ v = < 4, - 8 >$ $hat(v)=<4,-8>$

Let us see how we can add these two vectors:

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

Using component form:

$ˆ u + ˆ v = < 2, 5 > + < 4 - 8 >$ $hat (u) + hat (v) = <2 ,5 >+<4-8 >$

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

$ˆ u + ˆ v = < 2 + 4 > + < 5 - 8 >$ $hat (u) + hat (v) = <2+4>+<5-8 >$

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

We can represent this solution graphically as follows:

enter image source here

The solution is represented by

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

OR

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

Note: Alternative graphical solution process:

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

OR

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

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

Hope it helps.