Addition of Vectors by Rectangular Components

by • 04/07/2012 • GeneralComments (0)746

Addition of Vectors by Rectangular Components

Consider two vectors A1 and A2 making angles θ1 and θ2 with x-axis respectively as shown in figure. A1 and A2 are added by using head to tail rule to give the resultant vector A.

The addition of two vectors A1 and A2 mentioned in the above figure consists of following four steps.


Step 1

For the x-components of A, we add the x-components of A1 and A2 which are A1x and A2x. If the x-components of A is denoted by Ax then

Ax = A1x + A2x

Taking magnitudes only

Ax = A1x + A2x

Or

Ax = A1 cos θ1 + A2 cos θ2 …………….. (1)


Step 2

For the y-components of A, we add the y-components of A1 and A2 which are A1y and A2y. If the y-components of A is denoted by Ay then

Ay = A1y + A2y

Taking magnitudes only

Ay = A1y + A2y

Or

Ay = A1 sin θ1 + A2 sin θ2 …………….. (2)


Step 3

Substituting the value of Ax and Ay from equations (1) and (2) respectively in equation (3) below, we get the magnitude of the resultant A

A = |A| = √ (Ax)2 + (Ay)2 ……………… (3)


Step 4

By applying the trigonometric ratio of tangent θ on triangle OAB, we can find the direction of the resultant vector A i.e. angle θ which A makes with the positive x-axis.

tan θ = Ay / Ax

θ = tan-1 [Ay / Ax]

Here four cases arise

(a) If Ax and Ay are both positive, then

θ = tan-1 |Ay / Ax|
(b) If Ax is negative and Ay is positive, then

θ = 180º – tan-1 |Ay / Ax|
(c) If Ax is positive and Ay is negative, then

θ = 360º – tan-1 |Ay / Ax|
(d) If Ax and Ay are both negative, then

θ = 180º + tan-1 |Ay / Ax|

 

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