How do you find unit vectors that make an angle with another vector?

I need to find two unit vectors that make a 60 degree angle with the vector %26lt;3,4%26gt;. help?|||The important formula is the formula for dot products:





For any two real vectors v and w, we have





v dot w = |v|*|w|*cos(theta)





where theta is the angle between v and w. We have theta=60 degrees, so cos(theta)=1/2. Let us call %26lt;3.4%26gt; v and we want to find two possible w. We know that |w| is 1, because it is a unit vector, and therefore we are left with





%26lt;3, 4%26gt; dot %26lt;x, y%26gt; = 5/2





(|v|=5)





Thus, we are left with the equation





3x + 4y = 5/2





With the added condition that





x^2+y^2=1, because we want it to be a unit vector.





Solving the first equation yields





y=5/8 - 3x/4





Plugging this into the second yields





(5/8-3x/4)^2+x^2=1





25/64 - 15x/16 + 9x^2/16 +x^2=1





25x^2/16 - 15x/16 - 39/64=0





100x^2 - 60x - 39 = 0





Solving for x via the quadratic formula will yield two possible values. Then you can use the other two equations we developed to solve for y, and get two values for w, as desired.





=D|||ok, here's my try. you'll have to convert them to unit vectors because i forgot the function to do that.





1. %26lt;3, 0.58%26gt;


2. %26lt;-3, -3.45%26gt;





Basically, draw the vector. make a right triangle with the vector and the x axis. Use tangent to find the angle between the vector and the x axis (use tangent because you know the two legs, 3 and 4). I got 71 degrees. So two vectors 60 degrees from this would have central angle measures of 11 degrees and 131 degrees (which is actually 49 degrees) Use tangent again to find the unknown leg of the two traingle (which would be the y component of the vector), and you got the components of the vector. Again, the triangles are from the x axis. hope this helped, really.|||The angle 胃 of the given vector is:





胃 = arctan(4/3) 鈮?53.13010235掳





Now we need to find the tangents of the angles:





胃 + 60掳


胃 - 60掳





Let's look at the first one.





tan(胃 + 60掳) = (4/3 + 鈭?) / [1 - (4/3)(鈭?)]


tan(胃 + 60掳) = (4 + 3鈭?) / (3 - 4鈭?) = (-48 - 25鈭?) / 39


tan(胃 + 60掳) 鈮?-2.34105821





A vector with this tangent is:





v = %26lt;-1, 2.34105821%26gt;





|| v || = 鈭歔(-1)虏 + 2.34105821虏] = 2.545693136





Divide by its magnitude to get a unit vector v'.





v' = v / || v || = %26lt;-0.392820323, 0.9196152423%26gt;





The other vector can be calculated similarly.


______________





tan(胃 - 60掳) = (4/3 - 鈭?) / [1 + (4/3)(鈭?)]


tan(胃 - 60掳) = (4 - 3鈭?) / (3 + 4鈭?) = (-48 + 25鈭?) / 39


tan(胃 - 60掳) 鈮?-0.1204802516





A vector with this tangent is:





w = %26lt;1, -0.1204802516%26gt;





|| w || = 鈭歔1虏 + (-0.1204802516)虏] = 1.007231598





Divide by its magnitude to get a unit vector w'.





w' = w / || w || = %26lt;0.992820323, -0.1196152423%26gt;