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Question 3: What do rectangular components mean? Explain addition of vectors by rectangular components.

Answer

Rectangular components

Replacing a vector by two or more than two vectors is called the resolution of vectors. The vectors so obtained are called the components of the original vector. If the components of the vector are mutually perpendicular then they are called the rectangular components of the original vector.

Addition of vectors by rectangular components

Addition of vectors by rectangular components, sometimes called the analytical method of vector addition, is yet another method by which vectors can be added. This method gives more accurate results than the graphical or head-to-tail method because it is more mathematical and free from the possible errors of measurements and the never-perfect drawings.

Suppose we add two vectors Vector Aand Vector Bby the method of addition of rectangular components and let their resultant is vecr, which in the given figure,  makes an angle θ with the x-direction.

First, we add Vector Aand Vector B  geometrically. Place the tail of Vector Bat the tip of Vector Aand join the tail of Vector A  to the head of Vector B.

Resultant vector = OB =   vecr——-        (A)

Second, we add the same two vectors by rectangular components.addition by rectangular components
Draw perpendicular from the tip of  Vector Aon x-axis which meets at point C. Ax and Ay are the x- and y-components of Vector A.

x-component of Vector AAx = OC———–  (i)
y-component of Vector AAy = CA = DE —– (ii)

Since Ax and Ay are perpendicular to one another and their resultant is Vector A, therefore, they are rectangular components of Vector A.

Now draw perpendicular from point B on x-axis at D.
It is clear from the diagram and the rectangular symmetry of the figure that,
x-component of  Vector B = Bx = AE = CD ——–    (iii)
y-component of Vector B  = By = EB ———            (iv)
Since AE and EB are at right angle to each other and their resultant is Vector B; these are the rectangular components of Vector B.
Similarly, resolve vecr  in rectangular components by drawing perpendicular from point B on x-axis which meets at point D.
x-component of  vecr = Rx = OD ——— (v)
y-component of  vecr = Ry = DB ———- (vi)
OD and DB are perpendicular to each other, therefore, are rectangular components of vecr.
From the figure, it is clear that,
OD = OC + CD
Substituting values from (v), (i) and (iii) respectively,

Rx = Ax + Bx         ———–                   (vii)

And DB = DE + EB
Substituting values from (vi), (ii) and (iv),

Ry = Ay + By         ———–             (viii)
Rx and Ry are the rectangular components of vecr ,

Magnitude of vecr= Rx + Ry

Substituting from (vii) and (viii),
Magnitude of vecr = ( Ax + Bx) + (Ay + By)
Hence generalizing the result, when we add the vectors by their rectangular components, the x-component of the resultant vector is obtained by adding the x-components of the vectors to be added and the y-component is obtained by adding the y-components of all the vectors to be added.
Mathematically,

Rx = Ax + Bx + Cx +        ——-      (P)
Ry = Ay + By + Cy +        ——-      (Q)

Magnitude of vecr

Magnitude of the resultant vector vecr  is found by applying the Pythagorean Theorem to the right angled triangle OBD in the figure,

resultant

Direction of  vecr

Direction of the resultant vector is calculated by,

direction

Where Rx and Ry are calculated from equations (P) and (Q), respectively. Putting the values of Rx and Ry and applying the rules for finding θ in the Cartesian coordinates, the direction of the resultant vector can be found.

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