Question

a rabbit trying to escape a fox runs north for 4.0 m, darts northwest for 8.0 m, then drops 1.0 m down a hole into its burrow. what is the magnitude of the net displacement of the rabbit? show your work for credit.

Explanation:

Step1: Resolve vectors into components

Let the north - south direction be along the y - axis (north is positive y), east - west direction be along the x - axis (east is positive x) and up - down direction be along the z - axis (up is positive z).
The first displacement $\vec{d}_1=(0,4.0,0)$ m.
The second displacement $\vec{d}_2$ has magnitude $d_2 = 8.0$ m in the northwest direction. In component form, $d_{2x}=-8.0\sin45^{\circ}\text{ m}\approx - 5.66$ m and $d_{2y}=8.0\cos45^{\circ}\text{ m}\approx5.66$ m, $d_{2z} = 0$ m. So $\vec{d}_2=(-5.66,5.66,0)$ m.
The third displacement $\vec{d}_3=(0,0, - 1.0)$ m.

Step2: Find the net displacement vector

The net displacement vector $\vec{D}=\vec{d}_1+\vec{d}_2+\vec{d}_3$.
$D_x=0 - 5.66+0=-5.66$ m, $D_y=4.0 + 5.66+0 = 9.66$ m, $D_z=0+0 - 1.0=-1.0$ m. So $\vec{D}=(-5.66,9.66,-1.0)$ m.

Step3: Calculate the magnitude of the net displacement

The magnitude of a vector $\vec{D}=(D_x,D_y,D_z)$ is given by $|\vec{D}|=\sqrt{D_x^{2}+D_y^{2}+D_z^{2}}$.
$|\vec{D}|=\sqrt{(-5.66)^{2}+9.66^{2}+(-1.0)^{2}}=\sqrt{32.04+93.32+1.0}=\sqrt{126.36}\approx11.2$ m.

Answer:

$11.2$ m