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Chapter 5: Problem 86

Show that the dot product of two nonzero vectors is positive if the angle between the vectors is an acute angle and negative if the angle between the two vectors is an obtuse angle.

Short Answer

Expert verified
The dot product of two non-zero vectors is positive when the angle between them is acute because the cosine of an acute angle is positive. It is negative when the angle between them is obtuse, because the cosine of an obtuse angle is negative. We have concluded this based on the definition of the dot product and cosine values for acute and obtuse angles.

Step by step solution

01

Understanding series of angles

Recall that an acute angle is any angle less than 90 degrees (or \(\frac{\pi}{2}\) radians), and an obtuse angle is any angle between 90 degrees and 180 degrees (or between \(\frac{\pi}{2}\) and \(\pi\) radians). The cosine of an acute angle is positive, and the cosine of an obtuse angle is negative.
02

Applying cosine values to the Dot Product equation

Using the dot product equation \(\textbf{a} \cdot \textbf{b} = ||\textbf{a}|| ||\textbf{b}|| \cos(\theta)\). Substitute the cosine values we analyzed in the previous step, taking into account the magnitude of vectors ||\textbf{a}|| and ||\textbf{b}|| are always positive (since they represent magnitude or length).
03

Interpretation of dot product in terms of angle

If the angle \(\theta\) is acute then \(\cos(\theta)\) is positive, as a result, the dot product \(\textbf{a} \cdot \textbf{b}\) is positive. If the angle \(\theta\) is obtuse then \(\cos(\theta)\) is negative, as a result, the dot product \(\textbf{a} \cdot \textbf{b}\) is negative.

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