Math  /  Geometry

QuestionSeveral unit vectors r,s,t,u,n\vec{r}, \vec{s}, \vec{t}, \vec{u}, \vec{n}, and e\vec{e} in the xy-plane (not threedimensional 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. \square 1. ne\vec{n} \cdot \vec{e} ? ? ? ? \square ? \square ? ? \square
2. st\vec{s} \cdot \vec{t} (Click on graph to enlarge)

Studdy Solution
Use the geometric definition of the dot product:
1. For ne\vec{n} \cdot \vec{e}:
- Since the angle between n\vec{n} and e\vec{e} is acute, cosθ>0\cos \theta > 0. - Therefore, ne\vec{n} \cdot \vec{e} is positive.
2. For st\vec{s} \cdot \vec{t}:
- Since the angle between s\vec{s} and t\vec{t} is acute, cosθ>0\cos \theta > 0. - Therefore, st\vec{s} \cdot \vec{t} is positive.
The dot products are:
1. ne\vec{n} \cdot \vec{e} is positive.
2. st\vec{s} \cdot \vec{t} is positive.

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