Bearings and Vectors
Trig Refresher
\[ \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \]
\[
\sin\alpha = \frac{y}{r}
\]
\[
y = r\sin\alpha
\]
\[\qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \]
\[
\cos\alpha = \frac{x}{r}
\]
\[
x = r\cos\alpha
\]
The point \(P\) can be represented by the position vector:
\[
\vec{P} =
\begin{pmatrix}
r\cos\alpha \\
r\sin\alpha
\end{pmatrix}
\]
Written in unit vector form:
\[
\vec{P} = r\cos\theta\,\mathbf{i} + r\sin\theta\,\mathbf{j}
\]
Example
The journey of the crow from Linear motion with bearings is shown again below:
With angles marked in
Using vectors
\[
\mathbf{a}
=
\begin{pmatrix}
125\sin 75^\circ \\
125\cos 75^\circ
\end{pmatrix}
\qquad
\mathbf{b}
=
\begin{pmatrix}
-100\sin 40^\circ \\
-100\cos 40^\circ
\end{pmatrix}
\]
\[
\mathbf{c} = \mathbf{a} + \mathbf{b}
\]
\[
\mathbf{c}
=
\begin{pmatrix}
125\sin 75^\circ - 100\sin 40^\circ \\
125\cos 75^\circ - 100\cos 40^\circ
\end{pmatrix}
\]
\[
\mathbf{c}
=
\begin{pmatrix}
56.46196 \\
-44.252063
\end{pmatrix}
\]
\[
|\mathbf{c}|
= \sqrt{(56.46196)^2 + (-44.252063)^2}
\]
\[
|\mathbf{c}|
= \sqrt{5146.198066}
\]
\[
|\mathbf{c}| = 71.73\ \text{km}
\]
Using the scalar product to find the angle
\[
\mathbf{a}
=
\begin{pmatrix}
125\sin 75^\circ \\
125\cos 75^\circ
\end{pmatrix}
\]
\[
|\mathbf{a}|
= \sqrt{(125\sin 75^\circ)^2 + (125\cos 75^\circ)^2}
\]
\[
|\mathbf{a}| = \sqrt{15625}
\]
\[
|\mathbf{a}| = 125\ \text{km}
\]
\[
\mathbf{b}
=
\begin{pmatrix}
-100\sin 40^\circ \\
-100\cos 40^\circ
\end{pmatrix}
\]
\[
|\mathbf{b}|
= \sqrt{(-100\sin 40^\circ)^2 + (-100\cos 40^\circ)^2}
\]
\[
|\mathbf{b}| = \sqrt{10000}
\]
\[
|\mathbf{b}| = 100\ \text{km}
\]
\[
\mathbf{a}\cdot\mathbf{c}
= a_1 c_1 + a_2 c_2
\]
\[
\mathbf{a}\cdot\mathbf{c}
= 125\sin 75^\circ\,(125\sin 75^\circ - 100\sin 40^\circ)
\]
\[
\qquad\quad
+ 125\cos 75^\circ\,(125\cos 75^\circ - 100\cos 40^\circ)
\]
\[
= 6817.259 - 1431.6596
\]
\[
= 5385.5994
\]
\[
\cos\theta
= \frac{a_1 c_1 + a_2 c_2}{|\mathbf{a}|\,|\mathbf{c}|}
\]
\[
\cos\theta
= \frac{5385.5994}{125 \times 71.73}
\]
\[
\theta
= \cos^{-1}\!\left(\frac{5385.5994}{8966.25}\right)
\]
\[
\theta = 53.083^\circ
\]
\[
\theta = 53.1^\circ\ (1\text{ d.p.})
\]
Bearing
\[
\alpha = 53.1^\circ
\]
\[
\alpha = 15^\circ + x^\circ
\]
\[
53.1 = 15^\circ + x^\circ
\]
\[
53.1 - 15^\circ = x^\circ
\]
\[
x^\circ = 38.1^\circ
\]
\[
\text{bearing of crow from base }
38.1^\circ + 90^\circ = 128.1^\circ
\]
\[
\text{The supplement of this is }
180^\circ - 128.1^\circ = 51.9^\circ
\]
\[
\text{bearing for crow to fly}
\]
\[
360^\circ - 51.9^\circ = 308.1^\circ
\]
