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