dimensions. If α , β , and γ are the angles * between A and the x -, y -, and z -axes, respectively [Fig. . (d)], we have (d) A cos , A A cos , A A cos α β γ ( .16a) In general, we have ( .16b) The magnitude of vector A is ( .16c) A position vector r can be expressed as ( .
) where x, y , and z are the components of r along x-, y-, z- axes, respectively. . VECTOR ADDITION – ANALYTICAL METHOD Although the graphical method of adding vectors helps us in visualising the vectors and the resultant vector, it is sometimes tedious and has limited accuracy. It is much easier to add vectors by combining their respective components.
Consider two vectors A and B in x - y plane with components A x , A y and B x , B y : ( . ) Fig. . Answer Let OP and O Q represent the two vectors A and B making an angle θ (Fig.
. ). Then, using the parallelogram method of vector addition, OS represents the resultant vector R : R = A + B SN is normal to OP and PM is normal to OS . From the geometry of the figure, OS = ON + SN but ON = OP + PN = A + B cos θ SN = B sin θ OS = ( A + B cos θ ) + ( B sin θ ) or, R = A + B + AB cos θ 2AB cos θ ( .24a) In ∆ OSN, SN = OS sin α = R sin α , and in ∆ PSN, SN = PS sin θ = B sin θ Therefore, R sin α = B sin θ or, α ( .24b) Similarly, PM = A sin α = B sin β or, β α ( .24c) Combining Eqs.
( .24b) and ( .24c), we get β α ( .24d) Using Eq. ( .24d), we get: α ( .24e) where R is given by Eq. ( .24a). or, cos SN OP