Finding k such that u and v form {\pi}/3 Radians

[phosgene]

Homework Statement

Let u = (1,k) and v = (3,4). Find k such that the angle between u and v is ${\pi}/3$ radians.

Homework Equations

$u{\bullet}v=||u|| ||v|| cos{\theta}=x_{1}x_{2}+y_{1}y_{2}$

$||u||=\sqrt{x^2 + y^2}$

The Attempt at a Solution

Firstly I calculate the length of v and find an expression for the length of u:

$||u||=\sqrt{1 + k^2}$

$||v||=\sqrt{3^2 + 4^2}$
$||v||=5$

Then I find an expression for the dot product:

$u{\bullet}v=3+4k$

I plug my expressions for the dot product and lengths into the definition of the dot product, and set $\theta$ to $\pi/3$, giving me:

$3+4k=5cos{(\pi/3)}\sqrt{1+k^2}$

as $cos{(\pi/3)}=1/2$, I can substitute $cos{(\pi/3)}$ in my equation for 1/2, giving:

$3+4k=5/2\sqrt{1+k^2}$

I rearrange and expand brackets to get:

$6+8k=5\sqrt{1+k^2}$

I then square both sides to get rid of the square root, then expand brackets:

$(6+8k)^2=25+25k^2$
$36 + 96k + 64k^2=25+25k^2$

I move everything to one side:

$39k^2+96k+11=0$

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 $\pi/3$ 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 $\pi/3$. 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!

 

[tiny-tim]

hi phosgene!

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 $\pi/3$ 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 $\pi/3$. Have I done something wrong?

no, you did everything fine

however, when you squared both sides (which was correct), you automatically introduced an extra solution (for cos = -1/2, ie 2π/3), and you now need to check which of your two solutions is for π/3 !

(ie just check that the dot-product is positive)

 

[phosgene]

Thanks:) I just found out using google that the dot product is negative if the angle between the vectors is greater than 90 degrees. I had no idea that this was the case (it wasn't mentioned in the lectures). So it's crystal clear now. Thanks again!