- #1
phosgene
- 146
- 1
Homework Statement
Let u = (1,k) and v = (3,4). Find k such that the angle between u and v is [itex]{\pi}/3[/itex] radians.
Homework Equations
[itex]u{\bullet}v=||u|| ||v|| cos{\theta}=x_{1}x_{2}+y_{1}y_{2}[/itex]
[itex]||u||=\sqrt{x^2 + y^2}[/itex]
The Attempt at a Solution
Firstly I calculate the length of v and find an expression for the length of u:
[itex]||u||=\sqrt{1 + k^2}[/itex]
[itex]||v||=\sqrt{3^2 + 4^2}[/itex]
[itex]||v||=5[/itex]
Then I find an expression for the dot product:
[itex]u{\bullet}v=3+4k[/itex]
I plug my expressions for the dot product and lengths into the definition of the dot product, and set [itex]\theta[/itex] to [itex]\pi/3[/itex], giving me:
[itex]3+4k=5cos{(\pi/3)}\sqrt{1+k^2}[/itex]
as [itex]cos{(\pi/3)}=1/2[/itex], I can substitute [itex]cos{(\pi/3)}[/itex] in my equation for 1/2, giving:
[itex]3+4k=5/2\sqrt{1+k^2}[/itex]
I rearrange and expand brackets to get:
[itex]6+8k=5\sqrt{1+k^2}[/itex]
I then square both sides to get rid of the square root, then expand brackets:
[itex](6+8k)^2=25+25k^2[/itex]
[itex]36 + 96k + 64k^2=25+25k^2[/itex]
I move everything to one side:
[itex]39k^2+96k+11=0[/itex]
Using the quadratic formula, I get the answer that k= -2.341058209 or -0.1204802515. Clearly one of these is wrong (or both), as there can't be two angles in the same quadrant that make an angle of [itex]\pi/3[/itex] with a vector. But both of these values do satisfy the equation for the dot product of these two angles when the angle between them is [itex]\pi/3[/itex]. Have I done something wrong?
PS: I wasn't sure if this was the right forum, as both maths forums seem to be calculus-orientated. Sorry if it's not in the right place!