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

$$\boldsymbol u$$

$$\begin{bmatrix} 4.33 \\ 2.5 \end{bmatrix}$$

$$\boldsymbol v$$

$$5 \begin{bmatrix} c_\beta \\ s_\beta \end{bmatrix} \approx \begin{bmatrix} -4.33 \\ 2.5 \end{bmatrix}$$

$$\boldsymbol w$$

$$5\begin{bmatrix} c_\gamma \\ s_\gamma \end{bmatrix} \approx \begin{bmatrix} -4.83 \\ -1.29 \end{bmatrix}$$