Question

If u and v are both unit vectors that make an angle of 60 degrees to each other, calculate
a) |2u+3v|
b) |4u-5v|

The answer key says the correct answer for a) is 4.71 and the correct answer for b) is 3.22.

a)|2u+3v|^2=|2|^2+|3|^2-2|2||3|cos(60)=7; 7^.5=~2.65

b|4u-5v|^2=|4|^2+|5|^2-2|4||5|cos(60)=21; 21^.5=~4.58

What am I doing wrong?

Answer 1

You should draw the parallelograms. For (a), you found the short diagonal. What you should have used was the 120° angle formed by placing the start of 3v at the tip of 2u. That means that you should have used
|2|^2+|3|^2-2|2||3|cos(120)=13+6=19; √19=4.36
4.71 is wrong
To check, if you add the components, you get
w=2u+3v=(2,0)+(3cos60,3sin60)=(2,0)+(1.5,1.5√3)
take the length of that and you get |w|^2 = 3.5^2 + 2.25*3 = 19

Do a similar recalculation for part (b). Maybe they key is wrong there as well.