Adding Vectors by Components

Finally we will now see the use of components and unit vectors. Remember how we discussed adding vectors graphically using a ruler and protractor. A better method is with the use of components, because then we can get our answers by pure calculation. In Fig. 1.1 we have shown two vectors  and  added to form , but we have also indicated all the components.

FIGURE 1.1 Adding vectors by components.

By carefully looking at the figure you can see that

This is a very important result. Now let's back-track for a minute. When we write

you should say, ''Wait a minute! What does the + sign mean?" We are used to adding numbers such as 5 = 3 + 2, but in the above equation ,  and  are not numbers. They are these strange arrow-like objects called vectors which are ''add" by putting head-to-tail. We should really write

where  is a new type of ''addition", totally unlike adding numbers. However A_x, B_x, A_y, B_y, C_x, C_y are ordinary numbers and the + sign we used above does denote ordinary addition. Thus  actually means C_x = A_x + B_x and C_y = A_y + B_y. The statement  is really shorthand for two ordinary addition statements. Whenever anyone writes something like  it actually means two things, namely D_x = F_x+ E_x and D_y = F_y + E_y. All of this is much more obvious with the use of unit vectors. Write  and   and  . Now

is simply

and equating coefficients of  and  gives

and

مواضيع ذات صلة

أصول عدم الانعكاس

الانعكاس يولد الانعكاس

الاحتمالية في الميكانيكا الإحصائية

القابلية للانعكاس وعدم القابلية للانعكاس

نظرة في تعددية الفيزياء

الاختزال وعدم القابلية للانعكاس

الواقعية العلمية وتغيير النظريات

نقص الإثبات ومذهب الذرائعية ومذهب الواقعية

مشكلة التمييز: متى يكون المنهج علميا؟

المنهج العلمي: من الاستقراء إلى التكذيب وما بعده

مناهج العلوم وثمارها

!Parity is not conserved