High number_ Chapter 1 space analytic geometry and vector algebra_ Quantity product of vectors

One Definition of quantity product ：

  Given two vectors α and  β, Define their quantity product as  

   α·β = |α|· |β| cosφ ,    among  φ yes The angle between these two vectors .

The quantity product of two vectors is quantity .

Theorem 2  The relationship between vector perpendicularity and quantity product     

vector α And  β The necessary and sufficient condition for mutual perpendicularity is  α·β=0.     Specify that the null principle is perpendicular to any vector .

Two Coordinate representation of quantity product

Because vectors i, j, k It's all unit vectors , And perpendicular to each other , Easy to come by i·i = 1,  j·j = 1, k·k=1.

A vector α = {a₁ , a₂ , a₃},   β ={b₁, b₂ , b₃},

α  · β = a₁b₁ + a₂b₂ + a₃ b₃.    This is the coordinate representation of the quantity product .

From theorem 2 You know , α And  β The necessary and sufficient condition for mutual perpendicularity is  a₁b₁ + a₂b₂ + a₃ b₃ = 0.

Expressed by the coordinates of the quantity product , The coordinate representation of the cosine of the angle between two vectors can be derived . set up α  And  β The angle between them is φ, From the definition of quantity product and coordinate representation

 

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