In unit-vector notation, what is the sum of A = (5.4m)i + (1.4m)j and B = (-14.0m)i +(6.4m)j.What are (b) the magnitude and (c) the direction of A+B (relative to i) ?

Magnitude ##11,61m##, angle ##137,8^@## anticlockwise from unit vector ##hat i##

In rectangular form, we may simply add the corresponding components to get :

##vecA+vecB= (5,4-14) hat i + (1,4+6) hat j = -8,6 hat i + 7,8 hat j m ##

Hence the magnitude may be found from Pythagoras as ##|vecA + vecB|=sqrt((-8,6)^2+7,8^2) = 11,61 m##

The direction may be given by ##theta=tan^(-1)((7,8)/(-8,6))=-42,2^@##

Since tan is negative in both the 2nd and 4th quadrants, we see from the components of the resultant vector that we here in the 2nd quadrant. Hence a reference angle of 42,2 implies that the actual angle is 42,2 degrees above the negative x axis and hence in standard form it is ##180-42,2=137,8^@##

So in full vector form, ##vecA + vecB=11,61angle 137,8^@ m##