Question

Let $\hat{\mathbf{N}}$represent the direction horizontally north, $\widehat{\mathbf{NE}}$represent northeast (halfway between north and east), and so on. Each direction specification can be thought of as a unit vector. Rank from the largest to the smallest the following dot products. Note that zero is larger than a negative number. If two quantities are equal, display that fact in your ranking. (a) $\hat{\mathbf{N}}\mathbf{·}\hat{\mathbf{N}}$ (b)$\hat{\mathbf{N}}\mathbf{·}\widehat{\mathbf{NE}}$ (c) $\hat{\mathbf{N}}\mathbf{·}\hat{\mathbf{S}}$(d) $\hat{\mathbf{N}}\mathbf{·}\hat{\mathbf{E}}$(e)$\widehat{\mathbf{SE}}\mathbf{·}\hat{\mathbf{S}}$ .

Short answer

The ranking from the largest to the smallest dot products is $\hat{\mathrm{N}}·\hat{\mathrm{N}}>\hat{\mathrm{N}}·\widehat{\mathrm{NE}}=\widehat{\mathrm{SE}}·\hat{\mathrm{S}}>\hat{\mathrm{N}}·\hat{\mathrm{E}}>\hat{\mathrm{N}}·\hat{\mathrm{S}}$.

Step-by-step solution

Given information 

The value of unit vectors is $\left|\hat{\mathrm{N}}\right|=\left|\widehat{\mathrm{NE}}\right|=\left|\hat{\mathrm{S}}\right|=\left|\hat{\mathrm{E}}\right|=\left|\widehat{\mathrm{SE}}\right|=1$.

The angle between $\hat{\mathrm{N}}$and $\hat{\mathrm{N}}$is $0^0\mathrm{C}$.

The angle between $\hat{\mathrm{N}}$and $\widehat{\mathrm{NE}}$is $45°$.

The angle between $\hat{\mathrm{N}}$and $\hat{\mathrm{S}}$is $180°$.

The angle between $\hat{\mathrm{N}}$and $\hat{\mathrm{E}}$is $90°$.

The angle between $\widehat{\mathrm{SE}}$and $\hat{\mathrm{S}}$is $45°$.

Scalar product between two vectors

Consider two vectors $\overrightarrow{\mathrm{a}}$and $\overrightarrow{\mathrm{b}}$with angle $\theta$between them. The value of the scalar product of these two vectors is given below:

$\overrightarrow{\mathrm{a}}·\overrightarrow{\mathrm{b}}=\left|\overrightarrow{\mathrm{a}}\right|\left|\overrightarrow{\mathrm{b}}\right|\mathrm{cos\theta }$

The value of the scalar product between two vector quantities changes with the value of the angle between the vectors.

(a): Dot product between N and N 

Using the scalar product formula between two vectors, the value of $\hat{\mathrm{N}}·\hat{\mathrm{N}}$is as follows:

$\begin{array}{l}\hat{\mathrm{N}}·\hat{\mathrm{N}}=\left|\hat{\mathrm{N}}\right|\left|\hat{\mathrm{N}}\right|\cos 0°\\ \hat{\mathrm{N}}·\hat{\mathrm{N}}=\left(1\right)\left(1\right)\left(1\right)\\ \hat{\mathrm{N}}·\hat{\mathrm{N}}=1\end{array}$

(b): Dot product between N and NE

Similarly, the value of $\hat{\mathrm{N}}·\hat{\mathrm{E}}$can be calculated as follows:

$\begin{array}{l}\hat{\mathrm{N}}·\widehat{\mathrm{NE}}=\left|\hat{\mathrm{N}}\right|\left|\widehat{\mathrm{NE}}\right|\cos 45°\\ \hat{\mathrm{N}}·\widehat{\mathrm{NE}}=\left(1\right)\left(1\right)\left(\frac{1}{\sqrt{2}}\right)\\ \hat{\mathrm{N}}·\widehat{\mathrm{NE}}=\frac{1}{\sqrt{2}}\end{array}$

(c): Dot product between N and S

The value of$\hat{\mathrm{N}}·\hat{\mathrm{E}}$is given by the following:

$\begin{array}{l}\hat{\mathrm{N}}·\hat{\mathrm{E}}=\left|\hat{\mathrm{N}}\right|\left|\hat{\mathrm{E}}\right|\cos 90°\\ \hat{\mathrm{N}}·\hat{\mathrm{E}}=\left(1\right)\left(1\right)\left(0\right)\\ \hat{\mathrm{N}}·\hat{\mathrm{E}}=0\end{array}$

(e): Dot product between SE and S

The value of $\widehat{\mathrm{SE}}·\hat{\mathrm{S}}$is given by the following:

$\begin{array}{l}\widehat{\mathrm{SE}}·\hat{\mathrm{S}}=\left|\widehat{\mathrm{SE}}\right|\left|\hat{\mathrm{S}}\right|\cos 45°\\ \widehat{\mathrm{SE}}·\hat{\mathrm{S}}=\left(1\right)\left(1\right)\left(\frac{1}{\sqrt{2}}\right)\\ \widehat{\mathrm{SE}}·\hat{\mathrm{S}}=\frac{1}{\sqrt{2}}\end{array}$

Ranking the dot products

The ranking is given by the following, comparing all the values from parts (a), (b), (c), (d), and (e):

$\hat{\mathrm{N}}·\hat{\mathrm{N}}>\hat{\mathrm{N}}·\widehat{\mathrm{NE}}=\widehat{\mathrm{SE}}·\hat{\mathrm{S}}>\hat{\mathrm{N}}·\hat{\mathrm{E}}>\hat{\mathrm{N}}·\hat{\mathrm{S}}$

Hence, the ranking from the largest to the smallest dot products is$\hat{\mathrm{N}}·\hat{\mathrm{N}}>\hat{\mathrm{N}}·\widehat{\mathrm{NE}}=\widehat{\mathrm{SE}}·\hat{\mathrm{S}}>\hat{\mathrm{N}}·\hat{\mathrm{E}}>\hat{\mathrm{N}}·\hat{\mathrm{S}}$.