# vectors

#### veronica1999

##### Member
Show that there are at least two ways to calculate the angle formed by the vectors [cos 19, sin 19] and [cos 54, sin 54].

1) I can draw a unit circle and easily see that the angle is 35.

2) Change the values to decimals and use the law of cosines.
(Tried this but the calculation was a bit messy)

or use cos(theta) = uv/ lullvl

3) Use dot product and get the equation for
cos(54-19) = cos54cos19 + sin54sin19

Did my answers get the point of the problem?

#### Reckoner

##### Member
Did my answers get the point of the problem?
I think so. The two simplest ways that I can think of would be:

1. Recognize that $$\langle\cos\theta, \sin\theta\rangle$$ is a unit vector with direction $$\theta$$ relative to the positive $$x$$-axis. So $$\langle\cos19^\circ, \sin19^\circ\rangle$$ makes an angle of $$19^\circ$$ with the $$x$$-axis, and $$\langle\cos54^\circ, \sin54^\circ\rangle$$ makes an angle of $$54^\circ$$. Taking the difference between the two angles, we get $$35^\circ$$ as the angle between the two vectors.

2. Using the formula for the angle $$\theta$$ between two vectors $$\mathbf u$$ and $$\mathbf v$$ gives

$\cos\theta = \frac{\mathbf u\cdot\mathbf v}{\|\mathbf u\|\|\mathbf v\|}$

$\Rightarrow\cos\theta = \cos19^\circ\cos54^\circ + \sin19^\circ\sin54^\circ$

$\Rightarrow\cos\theta = \cos35^\circ$

So $$\theta = 35^\circ$$.

#### Ackbach

##### Indicium Physicus
Staff member
Another method would be to increase the dimension by one, and use the cross product. In three dimensions, the cross product and the dot product each give you a distinct way to compute the angle between two vectors.