What do you suppose happens when
the vectors are in opposite directions, such as (1, 0)^{T} and (-1, 0)^{T} ?

The magnitude of the dot product is negative.

Here is a sampling of **b _{u}** and the dot product with

Angle | b | Result | Picture |
---|---|---|---|

000° | (1.000, 0.000)^{T} | 1.000 | |

015° | (0.966, 0.259)^{T} | 0.966 | |

030° | (0.866, 0.500)^{T} | 0.866 | |

045° | (0.707, 0.707)^{T} | 0.707 | |

060° | (0.500, 0.866)^{T} | 0.500 | |

075° | (0.259, 0.966)^{T} | 0.259 | |

090° | (0.000, 1.000)^{T} | 0.000 | |

105° | (-0.500, 0.866)^{T} | -0.259 | |

120° | (-0.500, 0.866)^{T} | -0.500 | |

135° | (-0.707, 0.707)^{T} | -0.707 | |

150° | (-0.866, 0.500)^{T} | -0.866 | |

165° | (-0.966, 0.259)^{T} | -0.966 | |

180° | (-1.000, 0.000)^{T} | -1.000 |

The **b _{u}** in each case is the unit vector represented
by
(cos θ, sin θ )

What do you imagine is the range of values for the dot product of the two unit vectors,
**a _{u} · b_{u}** ?