Theory of Vectors 3
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The document discusses the theory of vectors, focusing on unit vectors, rectangular components, and direction cosines. It explains how to derive these concepts using vector notation and provides an illustration to find the unit vector of a given vector. Additional resources for further learning are also mentioned.
Theory of Vectors 3
- 1. Physics Helpline L K Satapathy Theory of Vectors 3
- 2. Physics Helpline L K Satapathy Theory of Vectors - 3 Unit vector : The unit vector in the direction of vector A is defined as A = A A A = A . A Given Vector = ( Its magnitude ) ( Unit vector in its direction ) Unit vectors in the direction of x , y and z axes = i , j and k (4N) i A force of 4N is acting parallel to the x-axis (6N) j A force of 6N is acting parallel to the y-axis (8N) k A force of 8N is acting parallel to the z-axis
- 3. Physics Helpline L K Satapathy Theory of Vectors - 3 Rectangular components of a vector O P A B C D E F X Y Z Given vector OP = R Rectangular components OA = BD = EP = CF = Rx = Rx i OB = AD = FP = CE = Ry = Ry j OC = AF = DP = BE = Rz = Rz k Its scalar projections OA along X-axis OB along Y-axis OC along Z-axis
- 4. Physics Helpline L K Satapathy Theory of Vectors - 3 Using Polygon Law, we get OP = OA + AD + DP R = Rx i + Ry j + Rz k OA AD O A B C D E F X Y Z P OD DP OD + DP = OP . . . (2)2 2 2 OA + AD = OD . . . (1) 2 2 2 (1) & (2) OA + AD + DP = OP2 2 22 R = Rx + Ry + Rz . . . (3) 2 2 2 2 OD in XY plane , DP parallel to z-axis
- 5. Physics Helpline L K Satapathy Theory of Vectors - 3 Direction Cosines In OAP , In OBP , In OCP , cos = OP OA R Rx = cos = OP OB R Ry = cos = OP OC R Rz = cos + cos + cos =2 2 2 Rx + Ry + Rz R 2 2 2 2 R R 2 2 = = 1 OAP = OBP = OCP = 90 A B C D E F P O
- 6. Physics Helpline L K Satapathy Theory of Vectors - 3 Illustration Q. Find the unit vector in the direction of the vector R = 3 i + 4 j + 12 k The given vector is R = 3 i + 4 j + 12 k Comparing with R = Rx i + Ry j + Rz k we get Rx = 3 Ry = 4 Rz = 12 = R = R R ( 3 i + 4 j + 12 k ) 13 1 Ans: R = Rx + Ry + Rz = 9 +16 + 144 = 169 R = 13 2 2 2 2
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