M.2.C

Given quantities:

\[\begin{split}\alpha = 30 ^\circ \\ \beta = 150 ^\circ \\ \gamma = 195 ^\circ\end{split}\]

Proceed as follows to learn about how to describe the direction of vectors.

Steps

1. Point Coordinates

Points on the Unit Circle

The coordinates of point A on the unit circle are:

\[\begin{split}\begin{bmatrix} x \\ y \end{bmatrix} &= \begin{bmatrix} c_\alpha \\ s_\alpha \end{bmatrix} \\ &= \begin{bmatrix} \cos 30^\circ \\ \sin 30^\circ \end{bmatrix} \\ &\approx \begin{bmatrix} 0.87 \\ 0.5 \end{bmatrix}\end{split}\]

Find the coordinates of the three points A, B and C on the unit circle. Fill in the following table - and round to \(0.01\).

Point	\(\begin{bmatrix} x \\ y \end{bmatrix}\)
A	\(\begin{bmatrix} 0.87 \\ 0.5 \end{bmatrix}\)
B	\(\begin{bmatrix} \ldots \\ \ldots \end{bmatrix}\)
C	\(\begin{bmatrix} \ldots \\ \ldots \end{bmatrix}\)

Solution

Point	\(\begin{bmatrix} x \\ y \end{bmatrix}\)
A	\(\begin{bmatrix} 0.87 \\ 0.5 \end{bmatrix}\)
B	\(\begin{bmatrix} c_\beta \\ s_\beta \end{bmatrix} \approx \begin{bmatrix} -0.87 \\ 0.5 \end{bmatrix}\)
C	\(\begin{bmatrix} c_\gamma \\ s_\gamma \end{bmatrix} \approx \begin{bmatrix} -0.97 \\ -0.26 \end{bmatrix}\)

2. Unit Vector Components

Components of a Unit Vectors

The \((x,y)\)-components of the unit vector \(\boldsymbol e_\alpha\) sind:

\[\begin{split}\begin{bmatrix} e_{\alpha x} \\ e_{\alpha y} \end{bmatrix} &= \begin{bmatrix} c_\alpha \\ s_\alpha \end{bmatrix} \\ &= \begin{bmatrix} \cos 30^\circ \\ \sin 30^\circ \end{bmatrix} \\ &\approx \begin{bmatrix} 0.87 \\ 0.5 \end{bmatrix}\end{split}\]

Find the \((x,y)\)-components of the three unit vectors \(\boldsymbol e_\alpha, \boldsymbol e_\beta\) and \(\boldsymbol e_\gamma\). Fill in the following table - and round to \(0.01\).

Unit Vector	\((x,y)\)-components
\(\boldsymbol e_\alpha\)	\(\begin{bmatrix} 0.87 \\ 0.5 \end{bmatrix}\)
\(\boldsymbol e_\beta\)	\(\begin{bmatrix} \ldots \\ \ldots \end{bmatrix}\)
\(\boldsymbol e_\gamma\)	\(\begin{bmatrix} \ldots \\ \ldots \end{bmatrix}\)

Solution

Unit Vector	\((x,y)\)-components
\(\boldsymbol e_\alpha\)	\(\begin{bmatrix} 0.87 \\ 0.5 \end{bmatrix}\)
\(\boldsymbol e_\beta\)	\(\begin{bmatrix} c_\beta \\ s_\beta \end{bmatrix} \approx \begin{bmatrix} -0.87 \\ 0.5 \end{bmatrix}\)
\(\boldsymbol e_\gamma\)	\(\begin{bmatrix} c_\gamma \\ s_\gamma \end{bmatrix} \approx \begin{bmatrix} -0.97 \\ -0.26 \end{bmatrix}\)

3. Vector Components

Components of a Vector

The \((x,y)\)-components of the vector \(\boldsymbol u\) are:

\[\begin{split}\begin{bmatrix} u_x \\ u_y \end{bmatrix} &= 5 \begin{bmatrix} c_\alpha \\ s_\alpha \end{bmatrix} \\ &= 5 \begin{bmatrix} \cos 30^\circ \\ \sin 30^\circ \end{bmatrix} \\ &\approx \begin{bmatrix} 4.33 \\ 2.5 \end{bmatrix}\end{split}\]

Find the \((x,y)\)-components of the three vectors \(\boldsymbol u, \boldsymbol v\) and \(\boldsymbol w\). Fill in the following table - and round to \(0.01\).

Vector	\((x,y)\)-components
\(\boldsymbol u\)	\(\begin{bmatrix} 4.33 \\ 2.5 \end{bmatrix}\)
\(\boldsymbol v\)	\(\begin{bmatrix}\ldots\\ \ldots \end{bmatrix}\)
\(\boldsymbol w\)	\(\begin{bmatrix}\ldots\\ \ldots \end{bmatrix}\)

Solution

Vector	\((x,y)\)-components
\(\boldsymbol u\)	\(\begin{bmatrix} 4.33 \\ 2.5 \end{bmatrix}\)
\(\boldsymbol v\)	\(5 \begin{bmatrix} c_\beta \\ s_\beta \end{bmatrix} \approx \begin{bmatrix} -4.33 \\ 2.5 \end{bmatrix}\)
\(\boldsymbol w\)	\(5\begin{bmatrix} c_\gamma \\ s_\gamma \end{bmatrix} \approx \begin{bmatrix} -4.83 \\ -1.29 \end{bmatrix}\)