Dot product: Difference between revisions
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In [[linear algebra]], the dot product of two vectors <math>\vec{v} = (v_1, \cdots, v_n), \vec{w} = (w_1, \cdots, w_n)</math> is defined to be <math>\vec{v}\cdot\vec{w} = \sum_{i=1}^n v_i w_i</math>. The dot product is [[linear|bilinear]]. | In [[linear algebra]], the dot product of two vectors <math>\vec{v} = (v_1, \cdots, v_n), \vec{w} = (w_1, \cdots, w_n)</math> is defined to be <math>\vec{v}\cdot\vec{w} = \sum_{i=1}^n v_i w_i</math>. The dot product is [[linear|bilinear]]. | ||
[[Category:Linear algebra]]{{stub}} | For two vectors <math>\vec{v}</math> and <math>\vec{w}</math>, we also have <math>\vec{v} \cdot \vec{w} = \left \Vert \vec{v} \right \Vert \left \Vert \vec{w} \right \Vert \cos \theta</math>, where <math>\theta</math> is the angle that <math>\vec{v}</math> and <math>\vec{w}</math> form with each other. In particular, for nonzero vectors <math>\vec{v}</math> and <math>\vec{w}</math>, we have <math>\vec{v} \cdot \vec{w} = 0</math> if and only if <math>\vec{v}</math> and <math>\vec{w}</math> are perpendicular, and <math>\vec{v} \cdot \vec{w} = \left \Vert \vec{v} \right \Vert \left \Vert \vec{w} \right \Vert</math> if and only if <math>\vec{v}</math> and <math>\vec{w}</math> are parallel, from which the identity <math>\vec{v} \cdot \vec{v} = \left \Vert \vec{v} \right \Vert ^2</math> follows. | ||
[[Category:Linear algebra]] | |||
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Latest revision as of 22:10, 19 February 2022
In linear algebra, the dot product of two vectors 
is defined to be 
. The dot product is bilinear.
For two vectors 
and 
, we also have 
, where 
is the angle that and form with each other. In particular, for nonzero vectors and , we have 
if and only if and are perpendicular, and 
if and only if and are parallel, from which the identity 
follows.
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