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Chapter 6: Problem 67

Describe how to find the angle between two vectors.

Short Answer

Expert verified
The angle between the two vectors can be found by calculating the dot product of the vectors, their magnitudes and then using these values in the formula for cos(\(\theta\)), followed by applying the inverse cosine function to get the actual angle in radians or degrees.

Step by step solution

01

Define the vectors and dot product

Let's say there are two vectors \(a\) and \(b\). Each vector has two components \(a=(a_{1},a_{2})\) and \(b=(b_{1},b_{2})\). The dot product of \(a\) and \(b\) can be defined as \(a \cdot b = a_{1} \cdot b_{1} + a_{2} \cdot b_{2}\).
02

Calculate the magnitudes of the vectors

The magnitude of a vector can be found using the formula \(\|a\| = \sqrt{a_{1}^{2} + a_{2}^{2}}\) and for \(b\) it is \(\|b\| = \sqrt{b_{1}^{2} + b_{2}^{2}}\).
03

Implement the values in the formula for finding the angle

The cosine formula for the angle \('\theta'\) between two vectors \(a\) and \(b\) is \(\cos(\theta) = \frac{a \cdot b}{\|a\|\|b\|}\). Substitute the calculated values of dot product, \(a \cdot b\), and magnitudes, \(\|a\|\) and \(\|b\|\), into this formula.
04

Find the angle

To find the angle \(\theta\) between the vectors, we need to apply inverse cosine function to the result of step 3. That is, \(\theta = \cos^{-1}\left(\frac{a \cdot b}{\|a\|\|b\|}\right)\)

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