Question

A bird flies 3.6 km due west and then 1.8 km due north. Another bird flies 1.8 km due west and 3.6 km due north. What is the angle between the net displacement vectors for the two birds?

Answer 1

Define unit vectors as follows:
$\hat{i}$ is in the eastern direction.
$\hat{j}$ is in the northern direction.

The position of the first bird is
$\vec{a} = -3.6 \, \hat{i} + 1.8 \, \hat{j}$

The position of the second bird is
$\vec{b} = - 1.8 \, \hat{i} + 3.6 \, \hat{j}$

Let θ = the angle between the net displacement vector for the two birds.
By definition,
$\vec{a} . \vec{b} = |a| |b| cos\theta \\\\ \theta = cos^{-1} ( \frac{\vec{a}.\vec{b}}{|a||b|} )$

$\vec{a}.\vec{b} = (-3.6)(-1.8)+(3.6)(1.8) = 12.96$
$|a| = \sqrt{3.24+12.96} =4.025$
Similarly,
|b| = 4.025

Therefore
$\theta = cos^{-1} \frac{12.96}{4.025^{2}} =36.9^{o}$

Answer:  36.9°