Question

several unit vectors \\( \vec{r}, \vec{s}, \vec{t}, \vec{u}, \vec{n} \\), and \\( \vec{e} \\) in the xy - plane (not three - dimensional space) are shown in the figure. using the geometric definition of the dot product, are the following dot products positive, negative, or zero? you may assume that angles that look the same are the same. 1. \\( \vec{r} \cdot \vec{u} \\) 2. \\( \vec{r} \cdot \vec{s} \\) 3. \\( \vec{t} \cdot \vec{u} \\) 4. \\( \vec{n} \cdot \vec{e} \\) 5. \\( \vec{n} \cdot \vec{t} \\) 6. \\( \vec{e} \cdot \vec{s} \\) 7. \\( \vec{e} \cdot \vec{r} \\) 8. \\( \vec{s} \cdot \vec{t} \\)

Explanation:

Step1: Recall the geometric definition of the dot product

The geometric definition of the dot product is \(\vec{a}\cdot\vec{b}=\vert\vec{a}\vert\vert\vec{b}\vert\cos\theta\), where \(\theta\) is the angle between the two vectors \(\vec{a}\) and \(\vec{b}\), and \(\vert\vec{a}\vert\) and \(\vert\vec{b}\vert\) are the magnitudes of the vectors. Since all vectors are unit vectors, \(\vert\vec{a}\vert = \vert\vec{b}\vert=1\), so \(\vec{a}\cdot\vec{b}=\cos\theta\).

• If \(0\leq\theta < 90^{\circ}\), then \(\cos\theta> 0\) (positive dot - product).
• If \(\theta = 90^{\circ}\), then \(\cos\theta = 0\) (zero dot - product).
• If \(90^{\circ}<\theta\leq180^{\circ}\), then \(\cos\theta<0\) (negative dot - product).

Step2: Analyze each pair of vectors

1. For \(\vec{r}\cdot\vec{u}\):

The angle between \(\vec{r}\) and \(\vec{u}\) is greater than \(90^{\circ}\). Using \(\vec{a}\cdot\vec{b}=\cos\theta\) with \(\vert\vec{r}\vert=\vert\vec{u}\vert = 1\), since \(\theta>90^{\circ}\), \(\cos\theta<0\).

1. For \(\vec{r}\cdot\vec{s}\):

The angle between \(\vec{r}\) and \(\vec{s}\) is less than \(90^{\circ}\). Using \(\vec{a}\cdot\vec{b}=\cos\theta\) with \(\vert\vec{r}\vert=\vert\vec{s}\vert = 1\), since \(\theta < 90^{\circ}\), \(\cos\theta>0\).

1. For \(\vec{t}\cdot\vec{u}\):

The angle between \(\vec{t}\) and \(\vec{u}\) is greater than \(90^{\circ}\). Using \(\vec{a}\cdot\vec{b}=\cos\theta\) with \(\vert\vec{t}\vert=\vert\vec{u}\vert = 1\), since \(\theta>90^{\circ}\), \(\cos\theta<0\).

1. For \(\vec{n}\cdot\vec{e}\):

The angle between \(\vec{n}\) and \(\vec{e}\) is greater than \(90^{\circ}\). Using \(\vec{a}\cdot\vec{b}=\cos\theta\) with \(\vert\vec{n}\vert=\vert\vec{e}\vert = 1\), since \(\theta>90^{\circ}\), \(\cos\theta<0\).

1. For \(\vec{n}\cdot\vec{t}\):

The angle between \(\vec{n}\) and \(\vec{t}\) is \(90^{\circ}\). Using \(\vec{a}\cdot\vec{b}=\cos\theta\) with \(\vert\vec{n}\vert=\vert\vec{t}\vert = 1\), since \(\theta = 90^{\circ}\), \(\cos\theta=0\).

1. For \(\vec{e}\cdot\vec{s}\):

The angle between \(\vec{e}\) and \(\vec{s}\) is \(90^{\circ}\). Using \(\vec{a}\cdot\vec{b}=\cos\theta\) with \(\vert\vec{e}\vert=\vert\vec{s}\vert = 1\), since \(\theta = 90^{\circ}\), \(\cos\theta=0\).

1. For \(\vec{e}\cdot\vec{r}\):

The angle between \(\vec{e}\) and \(\vec{r}\) is greater than \(90^{\circ}\). Using \(\vec{a}\cdot\vec{b}=\cos\theta\) with \(\vert\vec{e}\vert=\vert\vec{r}\vert = 1\), since \(\theta>90^{\circ}\), \(\cos\theta<0\).

1. For \(\vec{s}\cdot\vec{t}\):

The angle between \(\vec{s}\) and \(\vec{t}\) is less than \(90^{\circ}\). Using \(\vec{a}\cdot\vec{b}=\cos\theta\) with \(\vert\vec{s}\vert=\vert\vec{t}\vert = 1\), since \(\theta < 90^{\circ}\), \(\cos\theta>0\).

Answer:

1. Negative
2. Positive
3. Negative
4. Negative
5. Zero
6. Zero
7. Negative
8. Positive