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State and prove triangular law of vector addition?

CLASS - XI PHYSICS (Kinematics)

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CLASS - XI PHYSICS (Kinematics)

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OB’ (vector Q) and OA
(vector P) are to be added and they are drawn from O. Let the angles they make with the X-axis be ф2
and ф1
respectively. Draw the vector AB = Q =
OB from the arrow of vector P, parallel to OB’.

Now draw OB (vector R). We have to prove vector OA + vector AB = vector OB. Draw perpendiculars AF, AG from A and perpendiculars BE , BH from B onto the X and Y axes respectively.

Component of vector (OA) P along X-axis = OF = P Cos ф1

Component of vector (AB)Q along X-axis = -FE = -Q Cos (180- ф2) = Q Cos ф2

Net resultant component along X-axis = P Cos ф1 + Q Cos ф2 = OF – FE = OE.

Y component of vector P or OA = AF = GO

Y component of vector Q or AB = GH

Net resultant along the Y-axis = P Sin ф1 + Q Sin ф2 = GO+GH = OH

X-component of vector OB = OE = same as that for vector OA + AB.

Y-Component of vector OB = OH = same as for the vector OA + AB

Vectors OA + AB = vector OB

Now draw OB (vector R). We have to prove vector OA + vector AB = vector OB. Draw perpendiculars AF, AG from A and perpendiculars BE , BH from B onto the X and Y axes respectively.

Component of vector (OA) P along X-axis = OF = P Cos ф1

Component of vector (AB)Q along X-axis = -FE = -Q Cos (180- ф2) = Q Cos ф2

Net resultant component along X-axis = P Cos ф1 + Q Cos ф2 = OF – FE = OE.

Y component of vector P or OA = AF = GO

Y component of vector Q or AB = GH

Net resultant along the Y-axis = P Sin ф1 + Q Sin ф2 = GO+GH = OH

X-component of vector OB = OE = same as that for vector OA + AB.

Y-Component of vector OB = OH = same as for the vector OA + AB

Vectors OA + AB = vector OB