**Measuring Angles Between Vectors**

The dot product in disguise

**16.17** Identify the alternate dot product formula, which expresses a dot product in terms of the magnitude of the vectors.

According to Problem 16.1, given vectors **v** = <*a*, *b*> and **w** = <*c*, *d*>, the dot product is defined according to the following formula.

**v** · **w** = *ac* + *bd*

You can also calculate the dot product according to an alternate formula, which is based on the magnitudes of the vectors and the cosine of the angle θ between those vectors.

The alternate version is handy when you don’t know the components of the vectors.

*Note: In Problems 16.18–16.21,* v = *<0, 5> and* w = *<2, 2>.*

**16.18 ...**

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