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Vector resolution and addition

Zeenath Asks:

Sir please send me the answers with the reasons and steps

  1. the drawing on the left shows two vectors A and B, and the drawing on the right shows their components.

    The scalar components of these vectors are:

    When the vectors A and B are added, the resultant vector is R, so that RAB. What are the x– and y-components of R?
  2. the displacement vectors A and B, when added together, give the resultant vector R, so that RAB. Use the data in the drawing to find the magnitude R of the resultant vector and the angle q that it makes with the +x axis.

 

 

Answer

In the first question as you can see that the X components (Ax and Bx being equal and opposite will cancel each other and therefore the X com ponent of the resultant is zero.

The Y components will add up and therefore the Y component of resultant = 3.4+3.4 = 6.8 m and this itself is the magnitude of the resultant as the X components have canceled out.

Hnit for Answer to the second question

A Question from Electric Flux and Gauss Theorem

Zeenath asked:

How can you prove that dS in φ =E.dS is dS Cos θ if the area is tilted at an angle ? I need the mathematical Steps.

 

Ans;

Dear Zeenath,
Electric Flux is defined as the total no of field lines passing normal to the surface. So while calculating, we need to consider the area of the surface normal to the field lines only. That is why we take the dot product of E and ds, where ds is the area vector (not just the area: Area vector is a vector whose magnitude is equal to the area and dirtected normal to the surface.)
Then by definition of dot product,
dφ =E.dS = EdS cos θ
which gives the component of E and the component of area vector in the direction of E (When area vector is in the direction of E, the actual component of area is perpendicular to E)
Hope the matter is clear.
Mathew Abraham

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