photo.jpg

## 2(b).8. Parallelogram Law of Vector Addition It states that "if two vectors acting simultaneously at a point are represented in magnitude and direction by the two sides of a parallelogram drawn from a point, their resultant is given in magnitude and direction by the diagonal of the parallelogram...

## 2(b).8. Parallelogram Law of Vector Addition It states that "if two vectors acting simultaneously at a point are represented in magnitude and direction by the two sides of a parallelogram drawn from a point, their resultant is given in magnitude and direction by the diagonal of the parallelogram passing through that point." ### Resultant of vectors A and B, by parallelogram law Can be obtained by either of the following two methods: **(i) Graphical method:** Let A and B be the two vectors. Whose resultant is to be determined. Let these two vectors be the two lines OP and OQ drawn from Complete the parallelogram OPTQ, vectors as the two sides, Join OT. By parallelogram law of vectors, the diagonal OT gives the resultant of A and B. **(ii) Analytical method:** Let us represent A and B by the two adjacent sides OP and OQ of parallelogram OPTQ. Let 𝜃 be the smaller angle between A and B. Draw TS, a perpendicular, from T, upon OP produced. In right angled triangle OST, OT^2 = OS^2 + ST^2 Also ST ------- OP + PS = sin 𝜃 In triangle PST, ST ------- PS = tan 𝜃 PS = PT cos 𝜃 = B cos 𝜃) PT = sin 𝜃 ( ST = PT sin 𝜃 = B sin 𝜃) Therefore, vector B can also be represented by B sin 𝜃 and B cos 𝜃 Applying the triangle's law of vectors to OPT, we get OT = OP + PT or R =A+B In triangle OST, tan 𝜃 = ST ------- OS = B sin 𝜃 ------- A+B cos 𝜃 Equations (3) and (4) give the magnitude and direction of the resultant. ### Special cases **(i) When 𝜃 = 0°, cos 𝜃 = 1, sin 𝜃 = 0** Substituting for cos 𝜃 in equation (3) R^2 = A^2 + B^2 cos^2 𝜃 + 2 AB cos 𝜃 + B^2 sin^2 𝜃 = A^2 + B^2 (cos^2 𝜃 + sin^2 𝜃) + 2AB cos 𝜃 or R^2 = A^2 + B^2 + 2 AB cos 𝜃 or R = √A2+B2+2 AB cos 0 Let β be the angle made by R with A R = √A2 + B2 + 2 AB x 1 = √(A+B)^2 R = A + B (maximum) **(ii) When 𝜃 = 180°, cos 𝜃=-1 ** Substituting for cos 𝜃 in equation R^2 = √A2+B2 + 2 AB (-1) = √A2 + B2-2 AB R=A-B (minimum) **(iii) When 𝜃 = 90°, cos 𝜃=0** Substituting for cos 𝜃 R^2 = √A2+B2 + 2AB ---